Mathematical slices of Molecular Biology
A. Carbone and M. Gromov
September 25, 2002
1
Introduction
Molecular biology and genomics promise to deliver a full set of structures
needed to describe biological systems at the cellular level. Biomolecular techniques provide detailed (sometimes quantitative) information on the molecular functioning of the cell and allow systematic intervention into the cell
functions. The scientific paradigm is shifting from “understanding the biological structure” to “tracking macromolecules”, “manipulating and controlling the living cell”, “creating artificial biomolecular structures”, “modeling
fragments of cell dynamics” and “identifying signatures of such fragments”.
Mathematically speaking, the molecular biology encompasses the high dimensional set of parameters which projects to the lower dimensional space
of the classical phenomenological biology. The behavior of the system within
this low dimensional space rarely admits a self-contained quantitative description but on the molecular level such a description is feasible and entails
great predictive power. The high dimensional space has a distinctive formal
structure(s) that can be described, to some extent, in general terms without
touching upon finer points, thus rendering the schema of molecular biology
accessible to mathematicians.
We shall try to transmit our enthusiasm for nano-level 1 cell biology and
biomolecular techniques to the mathematical audience. With our limited
1
10Å = 1nm = 10−3 µm = 10−9 m is a convenient scale in molecular biology, since most
biomolecules are about 1-10 nanometers in diameter.
1
(to say the least) knowledge of the cell we try to condense the picture into
definite statements in order to focus the attention on particular issues and,
unavoidably, missing many biological details. We hope the reader will find
this exposition sufficiently entertaining and provocative.
Acknowledgments: Our fascination with DNA nanotechnology began
with a lecture by Ned Seeman in 1996 in New York, since when we have had
the pleasure of following the progress of his research. Bud Mishra taught
us the fundamentals of algorithmic and molecular aspects of genomics, and
convinced us by his experience that a mathematician can productively work
on biological problems. Sergei Mirkin projected the conceptual beauty of designing subtle experiments, encouraging us to think about molecular biology
from a mathematical angle. Alexander Gorban demonstrated the possibilities
and limitations of mathematical and computational modelling of biophysical
systems. We thank our seminar speakers for the inspiring expositions of their
experimental and theoretical work. Our special gratitude goes to François
Képès who generously spent many hours with us sharing his vaste knowledge
of molecular biology and his ideas on structural organisation of biological
systems. We are indebted to him and to Ned Seeman for suggesting many
improvements to the manuscript, and to Christophe Soulé for clarifying several points. After the manuscript was almost completed, we were privileged
to have a glimpse on Eric Westhof’s views on macromolecular geometry, but
delegating these to mathematicians goes beyond the scope of this article.
2
Crick’s dogma
The ancestral memory and the program running the cell are encoded in
DNA, a long chain built from 4 small molecular units. In a living cell, some
segments of DNA are copied or, as molecular biologists say, transcribed, by
means of proteins into shorter chains of similar molecules. These chains,
called mRNA’s, are further translated to polypeptide chains (proteins) made
of 20 amino-acids (another class of basic small molecules). In the course of
translation (or soon thereafter) proteins fold into compact three dimensional
conformations, and mutually interacting, make up the architecture (mainly
by self-assembly) and the dynamics (e.g. catalytic activity) of the cell. The
production of mRNA and proteins is accompanied by a continuous process
2
of degradation, which is less understood (especially for proteins) than the
synthesis. This schematic picture, properly annotated to be truly correct,
will be further refered to as Crick’s dogma.
3
Static and dynamic structures
The dynamics of the cell is a continuous flow of small molecules channeled by
the interaction with macromolecules: DNA, RNA and proteins. The behavior
of small molecules obeys the statistical rules of chemical kinetics, where,
in particular, the rate of reaction is proportional to some powers (usually
small integers) of concentrations 2 of the reactants. Sizeable amounts of
macromolecules might be described in a similar way. At the mesoscale (≈
10−100nm) however, macromolecules and macromolecular complexes appear
as individuals 3 , tiny mechanical contraptions, handling and sheparding small
molecules: enzymes controlling metabolic pathways and being themselves
switched on and off by small molecules, RNA polymerase synthesizing RNA
out of nucleotides using DNA as a template, ribosomes synthesizing proteins
from amino-acids with the help of tRNA, proteasomes selectively degrading
proteins, etc. Some components of these machines can be used outside the cell
for directing specific mesochemical processes (i.e. chemistry on the mesoscale
such as PCR and protein engineering) and the main challenge is to create new
mesochemical devices, comparable in structural complexity and specificity
to those used by the cell. The design and function of some mesochemical
machines are presented later in the paper.
DNA. This is a long (sometimes very long!) word in the four letters A, T, C
and G. These letters name the four nucleotides constituting DNA. The
molecule of each nucleotide is built out of the sugar-phosphate group and
2
If the channeling and the compartmentalization effects induced by macromolecules
reach the nanoscale, then the averaging implicit in the notion of “concentration” and
“ideal kinetics” becomes questionable. Also, some small molecules, important for cell
regulation, appear in low numbers (e.g. 10) in small prokaryotic cells and statistics should
be applied more carefully. In the presence of enzymes, the polylinearity of kinetics may
break down, as it happens in metabolic pathways.
3
The individuality of a molecule, manifested in its geometrical, dynamical and statistical “shape”, emerges in the interaction of atoms constituting the molecule. The variability
and specificity of features grows with the size of the molecule.
3
A
T
A
C
T
G
C
T
G
A
C
Figure 1: The four nucleotides (on the top) and a polynucleotide chain (on the
bottom). The (sequence of) arrows and circles represent the sugar-phosphate
backbone, while the palettes correspond to bases. The small white circle
represents the phosphate group which disactivates (by loosing part of it)
in the course of polymerization, and then it is depicted with a small black
circle. If we break a DNA sequence into individual nucleotides, these will be
disactivated and will be unable to make a chain again without an import of
energy.
T
G
A
C
Figure 2: Hydrogen bonds between nucleotides (dotted arrows). Observe
that nucleotides are antiparallel under the hydrogen bonding and that the AT
interaction is weaker than the CG interaction as produced by two hydrogen
bonds in AT and three bonds in CG. Each dotted arrow indicates a bond
issuing from an hydrogen atom of the base, as seen in Fig. 4.
4
Figure 3: A chemical representation of the Watson-Crick complementary
bases.
3'
O
CH3
O
P
4
5
O
3' 2'
1'
4'
N N1 A
3
1
2
2
H
O
5
4
O
O
C 6
N3 2 1
G 1N
3 2
O
N
7
5
4
A
3
1
N N
2
2' 3'
O
N
5
O
O
O CH2
5
4
3
T 16
2
1'
O
3
2
N N1 6 5
G 4
O
N
3
7
8
9
CH2
O
1'
4'
2' 3'
O
O
O
O
O
O
2
3' 2'
1'
4'
4'
2' 3'
O
4
C
1
O
O
P
P
6
P
4'
1'
O
O
CH2
6
CH2
O
O
N
8
9
O
O
6
O
O
O
5
4
7
8
9
3' 2'
1'
4'
O
N
O
P
CH2
H
3
P
3' 2'
1'
4'
O
4'
1'
3'
2'
O
CH2
O
P
O
O
O
4
CH2
O
O
O
CH2
7 8
9
6 5
T
6
O
5'
N
O
P
O
5'
3'
Figure 4: A chemical representation of a double stranded DNA. The two
sugar-phosphate backbones are bridged by coupled base pairs (where the
H’s of the hydrogen bonds are not indicated for the sake of space).
5
the base attached to it. The sugar-phosphate group is the same for all four
nucleotides, while the specificity is due to differences between the bases: A
(Adenine), T (Thymine), C (Cytosine) and G (Guanine). (See Figs. 1 and
4.)
The sugar-phosphate group is naturally polarized. Each nucleotide can be
covalently bound to another one with a phosphate group bridging between
two consecutive sugars, and this bond agrees with the polarization, as it
is schematically shown in Fig. 1, with a more realistic picture displayed in
Fig. 4. This process can be repeated almost indefinitely and produces long
chains of nucleotides called polynucleotides or single stranded DNA. Short
polynucleotides composed of 100 nucleotides are called oligonucleotides.
They can be nowadays synthesized chemically according to a given specification of letters.
Crick-Watson complementarity. The functioning of DNA, and thus all
life on earth, depends on the particular affinity between the bases A to T ,
and C to G, which comes from another kind of chemical bonding between
nucleotides called hydrogen bonds, which are about 10 times weaker than the
covalent bonding described above. See table in Fig. 6.
If we throw many copies of the nucleotides A, T, C and G into a suitable
solvant 4 , then soon they will predominantly appear in complementary pairs
AT and CG. 5 See Fig. 2. (The covalent sugar-phosphate bonds between
nucleotides do not form spontaneously.) Furthermore, suppose that we have
a solution of various polynucleotides in a tube. Whenever two inverse complementary subwords appear in two (possibly the same, but not adjacent in
the chain) polynucleotides, these eventually come close together and attach
to each other along these subwords 6 . (See Fig. 7.) The longer these words,
the stronger this attachment will be.
4
In pure water, one observes stacking rather than pairing of nucleotides.
In a pot, every base can pair with every other base, including itself; G and C will
preferentially pair together, and there are suggestions that A and T also will. The A-T
pairing however may not be Watson-Crick, as there are three other types of A-T pairs,
reverse Watson-Crick, and two other (Hoogsteen) mutually reversed pairings, which in the
absence of a backbone seem to be equally likely. There are more than 30 possible hydrogen
bound pairs between bases and also several base triplets. The latter occur in nature, but
in RNA (some tRNA) rather than in DNA.
6
The specificity of AT and GC binding is by far more pronounced in polynucleotides
than for monomers. Very roughly, this is due to the size matching: A and G are approximately twice as large than T and C, so that the pairs AT and CG fit nicely into the DNA
5
6
hydrogen atom
carbon C 12
nitrogen, oxygen atoms
sulfur, phosphorus atoms
covalent bonds (distances between nuclei)
hydrogen bonds
van der Waals radius
water molecule
sugars, amino-acids, nucleotides
globular proteins
ribosomes
viruses
bacteria
mitochondria
cell nucleus
animal cells
plant cells
DNA in a human cell
DNA in a human body
1.008Da
=def 12Da
14Da,16Da
32.06Da,30.97Da
0.75-2.3Å
1-2Å
2-3Å
3-4Å
0.5-1nm
2-10nm
25-30nm
26-60nm
0.5-5µm
2µm
3-10µm
10-30µm
10-100µm
2m
1014 m
18Da
150-500Da
5 · 103 -5 · 105 Da
2.5-4.5MDa
3-50MDa
5 · 103 -5 · 106 MDa
105 -106 MDa
106 -5 · 107 MDa
5 · 107 -5 · 109 MDa
5 · 107 -1011 MDa
5 · 106 MDa
200g
Figure 5: Table of linear scales and masses. All the numbers in the table are
approximate values. Da stands for 1 dalton, which is a unit of weight that
equals, by definition, 1/12 of the weight of an atom of C 12 . The convertion
constant “weight in daltons/weight in grams” is called Avogadro number NA .
This is the number of atoms in m grams of substance, where each atom
weights m daltons. The value of NA is ≈ 6.0221367 × 1023 , as found by
experiments. Thus, 1Da equals ≈ 1.66054 × 10−24 g. The weights of atoms in
the table are averaged over the distribution of the isotopes in the biosphere.
The size of an atom or of a molecule refers to the radius of the chemical
force it exerts in a particular class of chemical interactions (covalent, ionic,
van der Waals, etc.). Thus, the sizes of atoms can be regarded as distances
between nuclei of covalently bound atoms; these range between 0.75-2.3Å for
atoms present in the cell as ions or in molecules. Masses of macromolecules,
ribosomes and viruses refer to the “dry weight”, i.e. without water, while
larger unities are weighted with water, which makes ≈ 70% of the whole
weight. DNA makes about 1% of the
7 total weight in bacteria and about
0.25% in mammalian cells (cf. Alberts et al. which gives 200g in humans).
average kinetic energy of thermal motion per 0.9
molecule at room temperature (25◦ C ≈ 293◦ K)
energy of a hydrogen bond
1-5
approximate energy of covalent phosphate bond 50
between nucleotides
approximate energy of C = C bond
200
Figure 6: Energy table. The common chemical unit of energy is kcal/mol,
where 1mol is the amount of substance containing NA = 6.0221367 × 1023
molecules. For example, one mole of water makes about 18g, since the
molecule H2 O weights ≈ 18Da. Remind that 1kcal = 4, 184J, where 1 joule
(J) equals the kinetic energy of linear motion of 1kg of matter with the speed
of 1m/sec. The average thermal energy of linear motion of molecules at the
temperature T measured in Kelvins (K) (TKelvin ≈ TCelsius + 273), equals
3
kT joules, where k = 1.380658 × 10−23 JK −1 is the Boltzmann constant.
2
At the room temperature T = 298K, the thermal energy is 32 kT NA /4, 184
makes 0.8882855 ≈ 0.9kcal/mol as indicated in the table. The other energies in the table are also expressed in kcal/mol. The velocity V ofa particle
. For
of mass m kilograms with the kinetic energy T kelvins, is V = 3kT
m
−26
example, the water molecule ≈ 18Da ≈ 3 · 10 kg, has (quadratic) average
speed V = 640m/sec at room temperature.
T
C
G
A
C
T
T
A
G
T
C
G
G
T
C
C
A
A T C G G C A T C G G G A T A T C
A
T A G C C G T A G C C C T A T A G
C
C
T
A
A A G A A
G
T
T
G
C
T
A
G
A
C
G
A
C
T
A T
G C
C
G
A T
A T
G C
G
T
A
C
T G
T C
Figure 7: Hybridization of two strands of DNA (on the left) and a selfhybridization (on the right).
8
As molecules are in thermal motion, the binding along complementary
words, called hybridization, appears and disappears. There is a continuous
competition between hybridization of different complementary subwords in
the polynucleotides and there is no satisfactory quantitative theory predicting
the statistical distribution of hybridization. However, in accordance with
one’s intuition, the presence of pairs of long complementary subwords makes
the corresponding nucleotides stick along these subwords and stay together
for a long time, at least at the room temperature. As temperature reaches
roughly 100◦ C, the denaturation (i.e. the breaking of the hydrogen bonds
between the complementary nucleotides) becomes prevalent.
Association as well as dissociation of neighboring hydrogen bonds between complementary words are a positively correlated process leading to
a “zipping” effect: hybridization starts at the ends of two complementary
words and then persists at a high speed, or conversely, disengagement of two
words unzips from an end on. The (high dimensional) landscape of the energy describing this interaction appears as a multiscale network of rivulets
merging into the river corresponding to the zipping/unzipping mechanism.
The genetic information in a cell is encoded in a double stranded DNA,
consisting of two complementary strands of equal length held together (quite
strongly) by the hydrogen bonds. Thus, the unit of information is a pair
of complementary nucleotides, commonly abbreviated to bp for base pairs.
Fig. 8 gives approximate numbers of bps for genomes of various species.
In eukaryotic (i.e. “higher”, from yeast on) organisms 7 the genome is
organized into several units, called chromosomes, which are connected components or separate words of DNA, ranging in number from a few up to
several dozens. (See Fig. 8.) In bacteria there is typically a unique circular
DNA. Besides, there is extra genomic information contained in rather short
circular DNA present in both eukaryotic cells (in organelles) and in bacteria
(as plasmids).
A DNA word within a chromosome is intricately folded and spatially
(biologists say “topologically”) organized in a highly structured way, so that
a 4cm DNA for example fits into a chromosome body ≈ 3 − 10µm long
double helix. The actual spatial picture is described by Eric Westhof in his forthcoming
book on the stereochemistry of nucleic acids.
7
Eukaryotic cells are about 10 times larger than prokaryotic (i.e. bacterial) ones, and
they have a fine internal structure: a nucleus which contains DNA, a skeleton, semiindependent membrane-bounded entities called organelles, etc.
9
Species
Haploid genome
size
E. coli (bacteria)
4,640kb
S. cerevisae (yeast)
12,050kb
D. discoideum (slime mold) 70Mb
C. elegans (worm)
100Mb
A. thaliana (weed)
130Mb
D. melanogaster (fruit fly)
170Mb
Gallus domesticus (chicken) 1200Mb
M. musculus (mouse)
3Gb
X. laevis (toad)
3Gb
H. sapiens (human)
3Gb
Zea mays (maize)
5Gb
Allium cepa (onion)
15Gb
Haploid chromosome number
1 (circular)
16
7
11/12
5
4
39
20
18
23
10
8
Figure 8: The length of the genome in different species. The full set of genes
appears in several copies according to the species: once in bacteria and twice
in mammalians, for example. The haploid size refers to a single copy of the
set. The symbol kb denotes one thousand base pairs, Mb stands for 1000kb
and Gb denotes 1000Mb. The length of the genome only roughly corresponds
to the “complexity of the organism”. The organization of the genome into
chromosomes does not reflect the length of the genome.
10
and ≈ 1µm thick. In bacteria, the circular DNA is also compactified (by
supercoiling) but in a much simpler way than for eukaryotes.
The redundancy of having two strands instead of one, pays off in a variety
of ways:
replication: the replication mechanism using double stranded DNA
invented by nature seems to represent the simplest possible logical solution to the von Neumann problem of self-replicating automata, beating
von Neumann implementations by several orders of magnitude in simplicity and mathematical elegance. 8
repair: doubling of information allows the cell to employ various error
correction mechanisms to ensure the fidelity of replication.
stability: double stranded DNA is by far mechanically stronger and
chemically more resistant than the single stranded DNA.
Why four letters (A, T, C and G) rather than two? One can think of a
single stranded DNA as a word in the free group F2 generated by A, G with
T = A−1 and C = G−1 , where hybridization corresponds to cancellation
between reciprocal words. Four letters help to avoid excessive cancellation
(manifested by undesirable hybridization): two letters would make an abelian
(infinite cyclic) group.
Partial hybridization has more bio-chemical significance for RNA than for
DNA, as the latter appears in two strands “cancelling” each other and not
leading directly to sophisticated geometric patterns. The arrangement of the
ribosomal RNA (and, to a lesser extent, of transfer RNA) is reminiscent in
shape of the van Kampen diagrams familiar to combinatorial group theorists.
See Figs. 11 and 10. (One wonders whether the distribution of RNA/DNA
8
The traditional definition of automaton is unsuitable for expressing the idea of selfreplication. Von Neumann himself eventually modelled the problem in the framework of
cellular automata suggested by Ulam and the following developments, as far as we know,
have been limited to the Ulam-von Neumann setting. Intuitively, one wants to describe a
class (category, n-category?) of “structured systems” allowing high degree of complexity
interacting with the “environment” according to simple “rules”, such that in the course
of the dynamics new “identical” (or “similar”) systems appear. The problem is to replace
“” by specific definitions and then to decide whether a solution exists.
11
words in the abelianized group F2 /[F2 , F2 ] = Z2 or in the higher nilpotent
quotients of the free group bear any biological significance.)
RNA. Chemically, RNA is a twin sister of single stranded DNA: it has a
slightly different sugar-phosphate backbone and a small modification of the
four bases. It keeps the same A, C, G bases while T (Thymine) is replaced
by its chemical relative U (Uracil).
Unlike DNA, RNA appears in cells in millions of disconnected segments
100-1,000,000 nucleotides long. RNA’s are classified according to the function that they perform in the cell. Messenger RNA’s, in short mRNA, serve
as go-betweens DNA and proteins. Transfer RNA, or tRNA, and ribosomal
RNA, or rRNA, are incorporated into the machinery implementing the information carried by mRNA into synthesis of proteins. mRNA’s are 100-10,000
nucleotides long, tRNA’s are about 100 and rRNA are 2,000-5,000. Some of
these RNA’s are produced from short lived intermediates, called pre-RNA’s,
which can be up to 106 nucleotides long. In a rapidly growing mammalian
cell, the 80% of the total RNA is rRNA, the 15% is tRNA, while mRNA
make a small portion of the total RNA.
Why RNA at all? Could one design a cell where all functions of RNA
are performed by segments of single stranded DNA? The prevalent point of
view suggests that RNA appeared at the early stages of life with DNA and
proteins coming much later. This is witnessed by the presence of RNA in
several ancient machineries, including ribosomes and spliceosomes 9 . It was
discovered relatively recently that certain conformations of RNA (i.e. specific three dimensional shapes it takes under the influence of weak interactions
between its subsegments) display (auto-)catalytic properties similar to those
of certain enzymes (proteins), which can conceivably assist the replication
process. Thus RNA may appear in two distinct roles: a keeper of information and a worker executing this information for the purpose of replication.
Structurally, information is organized in a straightforward way as a string
of letters coding nucleotides, while the catalytic activity depends on a three
dimensional arrangement of the polynucleotide chain.
How does the linear chain of nucleotides acquire a specific three dimen9
See Section 3 for the definition of ribosome. Spliceosomes are ribonucleoprotein complexes roughly the size of ribosomes. They assist transformation of pre-mRNA to mRNA
in eukariotes.
12
sional shape? Its behavior in solution follows paths that can be rather faithfully described by a system of stochastic differential equations where the underlying potential U incorporates the interaction between various molecules
constituting the chain and those in the solution, and where the major contribution to U comes from the hydrogen bonds responsible for the complementarity. A computationally feasible solution to the system runs into a
multitude of serious problems (similar, but possibly less severe, than those
for protein folding) where the major difficulty is the exponential multitude
of the local minima of U . Some of these can be accounted for by the combinatorial patterns of matching between complementary words. This matching
determines what is called, the secondary structure of RNA, and there are
heuristic algorithms for its determination. Deriving the final three dimensional shape from the secondary structure remains an unsolved (to some
extent mathematical) problem.
Transcription is a transformation of specific segments of DNA into single
strands of RNA. There are certain proteins, called transcription factors, that
choose particular segments of DNA, which then serve as templates for the
production of the complementary segments of RNA. The synthesis is performed with the help of another protein, the RNA polymerase (which binds
to one of the strands of the double stranded DNA), with an average rate of
60 nucleotides per second in E. coli. (This number may be different for other
organisms.)
The resulting RNAs are further modified (cleaved and /or spliced) with
the help of various enzymes. For example, in eukaryotic cells, pre-mRNA is
spliced by cutting away several (possibly large) subsegments (introns) and
by sewing (ligating) together the remaining segments (exons).
The splicing process is not unique (even the division into introns and
exons is not canonical) but depends on a particular biochemical (physiological) state of the cell. Also, in viruses and prokaryotes, one segment of DNA
may be transcribed into to several disjoint segments of RNA. The “edited”
RNA is called messenger RNA, or mRNA. The length of mRNA produced
in a single cell may vary from few hundreds to 10,000 (and sometimes more)
nucleotides.
Pre-mRNA folds in the course of transcription and this folding plays a
role in the editing of pre-mRNA to mRNA. The co-transcriptional folding
follows a different path from that of free molecules in solution, due to the
13
constraints imposed by the transcription process. Roughly speaking, instead
of a single system we have a sequence of “correlated” systems of differential
equations time-indexed by the number of nucleotides transcribed at a given
moment, and depending on the composition of the synthesized segment. The
solution of these equations must be time-correlated with the time-indexing
of these equations.
Definition of a gene. Originally a gene was understood as an abstract
unit of heredity but there is no consensus nowadays on which structure or
biological function corresponds to such a unit on a molecular level.
As mathematicians we are less sensitive to the distinction between (spatial) “structures” and “functions”, as the latter can be sometimes seen as
structures in the space-time. With this in mind, we define10 a gene as a segment of DNA together with a transformation to a segment(s) of mRNA (or
other functional RNAs such as tRNA, or rRNA). This definition is supposed
to capture two phenomena: alternative splicing in eukaryotic cells, where the
same segment of DNA may lead to the production of different mRNA depending on a particular global state of the cell or on what happening in the
vicinity of the transcription site; overlapping genes, where different segments
of RNA are produced from overlapping segments of DNA, as it happens in
viruses and prokaryotes.
The first level of the structure of the genome consists of the sequences
of letters A, T, C, G with distinguished segments supporting genes. Parts of
these are transcribed to mRNA (with the introns spliced away) and other
parts, called regulatory region(s), are responsible on when and how the transcription takes place. (A regulatory region may be separated from the transcribed part by a long stretch of DNA possibly containing other regulatory
and transcribed segments. Sometimes such a region is respectfully called
regulatory gene.)
The intergenic space constitutes the bulk of DNA in most eukaryotes and
it is unrespectfully referred to as junk DNA; a similar disrespect is extended
to introns. See Fig. 9.
10
In biology, a definition is supposed to isolate a pronounced phenomenon but not
necessarily capture it completely. For instance, one cannot give a definition of a gene being
both concise and exhaustive. Besides, the mathematical standpoint suggests a definition
of an object along with the full category of related objects (and morphisms). Here the
relevant category can be made of genetic networks, so that a definition of gene would
include the regulation of the expression of the gene.
14
Species
E. coli (bacterium)
S. cerevisae (yeast)
C. elegans (worm)
D. melanogaster (fruit fly)
H. sapiens (human)
Protopterus (lungfish)
A. thaliana (plant)
Fritillaria (plant)
Genome size
4,640kb
12,050kb
100Mb
170Mb
3Gb
140Gb
130Mb
130Gb
% coding DNA
100
70
20
20
1
0.02
30
0.02
Figure 9: The length of the genome in different species compared with its
coding part, where the latter is a very rough and disputed estimate for large
genomes. According to the February 2001 news release, humans have about
30,000 genes. Taking 1000bps as an average length of the code for proteins,
one estimates the coding part of the human genome as 1%. (This percentage
goes up if the coding is understood in a more generous way, thus reaching
100% for bacteria.) We computed the other values in the table in a similar
way for the multicellular organisms.
15
The sequences of letters constituting different parts of DNA, both in
genes and in junk DNA, can be thought of as random sequences of intricately
correlated letters 11 . These correlations within the transcribed regions reflect
the structure and function of proteins eventually produced from mRNA, while
the intergenic space is organized under the influence of several competing
factors: melting and bending properties of the DNA helix, genetic parasites
such as self-splicing introns, neutral mutations, etc. where the full list of
factors is unknown.
The spatial organization of DNA in chromosomes is essential for replication and transcription. For example, two segments which are far in the
sequence but close in space (because of the folding) may be targeted by the
same transcription factor.
Proteins. Proteins are polymers made of 20 basic amino-acids joined by
peptide bonds. The length of natural proteins varies from a few dozens to
several thousands amino-acids with a typical value of 200-300 amino-acids.
An amino-acid is a compound consisting of two parts: a constant part formed
by an amino group, a carboxyl group and a hydrogen atom (ACH), and a
variable part, called side chain, which comes (in the existing organisms) in
20 flavors. Thus, a linear protein is a word in 20 letters represented by 20
amino-acids, where consecutive amino-acids are joined by peptide bonds at
their ACH groups.
There is a formal similarity between polynucleotides and amino-acids.
Both are heteropolymers, i.e. assemblages of several kinds of standard molecules,
called monomers, having a connected chemically homogeneous (topologically
linear) backbone, with chemically different short branches attached to each
monomer of the backbone.
Different side chains have specific mutual interactions via weak bonds12 ,
which force a linear protein to fold in solution (under suitable temperature
11
The level of correlations in the prokariotic DNA is comparable to that in the text of
the “Hamlet”. Eukaryotic DNA has a large component of uncorrelated white noise due to
neutral mutations in the non-coding regions.
12
The essential weak bonds in folded proteins are hydrogen bonds, ionic bonds (in watersolutions; in crystals, for instance, these can be as strong as covalent bonds), van der Waals
interactions and hydrophobic bonds (we return to the latter in the section on Liposomes
and minimal surfaces). Weak bonds are sometimes reinforced by covalent S-S bonds
between neighboring cysteine residues in a folded polypeptide chain. Cysteine (one of the
two amino-acids containing sulphur) ensures the stability of snake poisons for example.
16
and acidity) into a particular geometric shape(s). There is a whole field
dedicated to how and why proteins fold, with about 10,000 instances where
the three-dimensional conformation (folding) is known (by a variety of biophysical, bio-chemical and bio-informatical techniques). Yet, there is no clear
conceptual picture on the nature, unicity and origin of the folding.
Roughly, as for RNA, the folding of a polypeptide chain comes as a solution of a system of stochastic differential equations where the final spatial
conformation of the protein represents the minimum of the (free) energy
of the molecule, incorporating weak bonding, bending and torsion energies,
etc. A direct solution of this problem is computationally unfeasible and, in
practice, one resorts to comparing a given protein to a similar one(s) where
the structure is already known by using pattern matching/perturbation techniques. The true problem however, is not so much finding the conformation
but identifying the active sites, that are specific locations on the external
part of the three-dimensional conformation. An active site has a particular
geometry and energy profile responsible for the enzymatic/binding activities
of the protein, and involves one or several short peptide subchains of the
protein. Most often, active sites are crevices of the protein accessible from
the outside.
The “inverse folding problem”, that is finding a word written in the 20
letters alphabet where the spatial conformation of the corresponding protein
displays a site with a required property, lies at the core of the protein design,
e.g. in pharmacology.
Proteins, carrying out the “program encoded by the genes” 13 , run around
the cell (or cell’s compartments) and interact with other molecules: these
can be other proteins and peptides (i.e. leftovers of partial degradation of
proteins), (poly)nucleotides, lipids, polysaccharides and about 800 various
kinds of small molecules. A large group of proteins called enzymes accelerate
a variety of chemical reactions, other proteins make cell’s architecture such
as membranes and cytoskeleton, a third group regulates gene transcription,
a fourth group transports molecules across the cell, yet another group is
responsible for extracellular communication, etc.
Translation from mRNA to proteins. The formal aspect of the translation
13
This politically loaded widespread metaphor has the same kind of purposes and limitations as “the fittest survives” and ranks below “one gene −→ one mRNA −→ one protein
−→ one function”.
17
is quite easy: each amino-acid is coded by a triplet of bps called codon.
As there are 64 such triplets, non surprisingly some amino-acids are coded
by different codons. The translation begins with the recognition of a start
codon, usually AUG, which determines the reading frame, i.e. a division of
the mRNA chain modulo 3, and continues until a stop codon is found, which
is usually either UGA, UAA, or UAG.
The codon–amino-acid correspondence is identical for most known organisms in agreement with the hypothesis that life on earth descends from
a single macromolecular complex 14 . (Exceptions: many mitochondria and
some protozoans differ in a few codons from the rest of organisms on earth.
This is apparently due to the later evolutionary development.)
The actual translation from mRNA to proteins is a highly sophisticated
mesochemical process (which can be reproduced in vitro albeit by far less
efficiently than in the living cells). This is performed by two groups of macromolecular complexes, ribosomes and transfer RNA.
Ribosomes are roughly spherical, protein-synthesizing machines. They
are huge by molecular standards, up to 30nm in diameter and about 3-10
times heavier than an average mRNA. They are composed of several different
ribosomal RNA molecules (rRNA) making about 2/3 of their weight, and of
more than 50 proteins. Fig. 10 represents an example of rRNA.
A transfer RNA (for short tRNA) is ≈ 80 nucleotides long and folds
in a particular way in obedience to the Crick-Watson complementarity 15 .
It displays an anti-codon at a specific site somewhere in the middle of the
strand and has an amino-acid attached to it (see Fig. 11). This complex is
called aminoacyl-tRNA. The anti-codon is the triplet complementary to the
one which encodes the amino-acid attached to this tRNA. The linkage between tRNAs and the corresponding amino-acids is performed by enzymes,
called aaRS. There are exactly 20 of these (of size ranging from 500 to several
thousands amino-acid residues) corresponding to the 20 amino-acids. Each
aaRS brings together the tRNA and the amino-acid coded by the tRNA. The
14
A molecule is a collection of atoms joined together by strong inter-atomic forces. A
molecular complex is a conglomeration of several molecules kept together by weak (e.g.
hydrogen bond) forces. With some stretch of imagination, a cell can be regarded as such a
complex. Molecular complexes are regarded as dynamic rather than static entities, where
the dynamics of the cell is structurally different from that of purely chemical complexes.
15
The spatial (tertiary) conformation of tRNA, and RNA in general, depends on more
factors than sheer complementarity.
18
32 30
33
34
20
21
28
35
31
29
P19
19
P21-2
P21-3
17
P18 P21-1
1
16
27
26
23
24
25
36
2
38
3
15
37
5'
4
14
5
13
12
40
6
3'
39
7
11
10
9
8
10 nucleotides
Figure 10: Secondary structure (planar representation of an energetically
significant part of the folding) of prokaryotic ribosomal RNA.
19
5'
D G A
U G C G
C
G
G C G C
G D A
U
C
U
C
C
C
A
C 3'
Amino acid
C
(alanine)
A
C
G
C
G
G
U
C
G
G
C
U
U
G C
U U
A
A G G C C
G
U C C G G
C
G
T
Ψ
A C
A
D
G
G
G
G
G
Ψ ml
Anticodon loop U
U 1 2 3
I G C
Anticodon
C C G
3'
3
2
1
Codon
5'
mRNA
Figure 11: Secondary structure of the transfer RNA for alanine (one of the
amino-acids) consisting of 76 nucleotides. The shaded letter I, for InoSine,
in the anticodon, is a A lacking an aminogroup (i.e. N H2 −C≡ is replaced by
H−C≡). This facilitates the non standard pairings C-I and A-I, and allows
the cell to use fewer tRNA’s than there are codons. DNA of all organisms
contain 61 codons; bacteria have 30-40 tRNA’s, and animal mitochondria
manage with 22. On the other hand, animals and plants have 50-100.
20
Figure 12: Schema of a small fragment of the metabolic pathway. (The
direction of the reactions is not indicated.)
recognition of the appropriate tRNA by an aaRS depends on particular motives in the tRNA sequence, with a significant role played by the anti-codon.
The pairing tRNA/amino-acid is the first major step of translation. The
second step is the synthesis of the protein: very roughly, tRNAs get attached
to an mRNA, the anticodon to the corresponding codon, thus aligning the
amino-acids attached to these tRNAs. The latter are sewed together to a
protein by a ribosome, at an average rate of 40 amino-acids per second for
E. coli (and differs from organism to organism).
Metabolic pathways and genetic networks. A cell is a chemical machine
in constant interaction with the environment. It takes from the outside
nutrients 16 and oxygen (unless it is anaerobic) which are transformed into
16
The major source of energy for plant cells is not external chemicals but rather quanta
of visible light which are transformed to chemical energy via photosynthesis. The sites
of photosynthesis in plant cells and green algae are chloroplasts. These are independent
entities, organelles, up to 10µm long and typically 0.5–2µm thick. In many respects, they
are similar to mitochondria in animal cells. They both contain their own DNA and the
21
energy 17 that fuels the cellular machinery: transcription, translation and
replication.
The processing of chemicals is organized into metabolic pathways. Schematically one might think of a directed graph M where the edges are marked
by various chemical compounds, usually small molecules, and where vertices
represent chemical transformations. (For an example, see Fig. 12) The direction of the flow in the graph indicates the predominant direction of the
reactions reflecting the decrease of the free energy. Left to themselves, the
chemicals in a cell would react extremely slowly. The rate of reaction are
many fold times enhanced by high local concentrations within the walls of
the cell architecture, and by catalytic effects of relevant enzymes present at
the sites of the reactions 18 .
Metabolic pathways are regulated by changing enzyme activity via positive and negative feedback loops. This allows the cell to sustain homeostasis,
that is the (almost) constant level of the final product in the variable environment. In many situations, the first enzyme of a chain of reactions is
inhibited by a negative feedback effect of the final product of the pathway.
The inhibition is achieved by the binding of the final product to the enzyme,
proteins encoded by these are synthesized within their organelles. However most of the
proteins in each organelles are encoded in nuclear DNA.
17
Most of this energy is carried by small molecules, called ATP, consisting of adenine
(the base A) joined with ribose (the sugar making the backbone of RNA), together with
three phosphate groups. When ATP hydrolyzes (i.e. reacts with water), it releases energy
of ≈ 12kcal/mol in the chemical environment of the cell. The hydrolysis of ATP has a
non-negligible activation energy and does not occur spontaneously without the presence
of a catalyser. Enzymes serve as such catalyzers: they employ the energy of ATP to drive
uphill reactions which need an import of free energy.
18
The number of small molecules involved might be quite large, their diffusion in cytoplasm is as fast as in water (≈ 0.1sec to go across a cell of 10µm), and their chemistry
is governed, up to some extent, by the polylinear rules of the ideal kinetics: at a given
temperature the percentage of chemicals being transformed to the final product is proportional to (certain powers of) their concentration; enzymes cannot change the direction
of a chemical reaction but only the speed at which the process approaches equilibrium,
i.e. the state where the rate of direct and inverse reactions mutually cancel each other.
Besides, enzymes can promote energetically unfavorable reactions, for instance making a
covalent bond between X and Y , by bringing ATP’s (or other molecules carrying free
energy like GTP) to the site of the reaction and by coupling the energy released by the
ATP hydrolysis to the reaction between X and Y . Without an enzyme, such a process
would be highly unlikely as it needs a meeting of three molecules X, Y and ATP, properly
positioned with respect to each other.
22
where the binding process is not ruled by ideal kinetics, but depends on
the particular combinatorial arrangement of enzymes into complexes. This
produces a highly non-linear dependence of binding on concentration with a
pronounced threshold effect where a small excess of the final product may
lead to an almost complete inhibition of the enzymatic activity of the protein.
(This is called collective allosteric transition of proteins.)
The metabolic self-regulation process has a fast response time and is
easily reversible. More radical regulation consists in changing the rate of
production and degradation of enzymes, which is achieved by suppressing
and/or activating the transcription of relevant genes. In fact, there is a
second network, the gene regulatory network, responsible for this process,
which is achieved with regulatory proteins binding to the regulatory regions
of the relevant genes.
Roughly, a genetic network is a directed graph G whose nodes represent
the genes. An arrow issued from a vertex is marked by the protein 19 coded
by this gene, while incoming arrows tell us that the transcription of the
gene is influenced by the protein. The current studies exhibit subgraphs
of this graph with outcoming degree up to 100 and incoming degree up to
15. The most common motive in the network of E. coli is a simple negative
feedback loop 20 , by which a gene sustains a constant level of activity. More
complicated patterns are being found as well.
A large amount of outcoming arrows is labelled by enzymes and ends up
in the nodes of the graph of metabolic pathways. Some other arrows correspond to functional proteins involved into structure, transport, bio-chemical
signalling, etc.
There is an extra combinatorial structure to the graph M ∪ G expressing
the effects of small molecules on enzymes and regulatory proteins. The first
example of the latter was discovered by Jacob and Monod in 1961, who
found out that if the nutrient glucose is replaced by a somewhat less tasty
lactose, the bacteria E. coli starts producing more enzymes needed for the
assimilation of lactose (there are three of these enzymes). Normally these
enzymes are produced by genes which are suppressed by a certain regulatory
protein. The presence of significant amount of lactose in the cell disactivates
19
Our definition of a gene specifies an mRNA and thus the corresponding protein.
Such a motive may turn out to be common for fast growing bacteria, but this is not
the case for the fast growing eukaryote yeast, for instance.
20
23
this protein, and the lactose digesting enzymes are produced.
One can think of this auxiliary structure as a family of subgraphs in
G parametrized by various bio-chemical and physical conditions of the cell,
e.g. concentration of particular small molecules, temperature, acidity, etc.
Each point in the parameter space distinguishes a relatively small subgraph
indicating the part of the network active under a given condition.
The major mechanism of regulation consists in binding of proteins to
regulatory regions which enhances or inhibits the transcription. We think
of the binding as a (stochastic) mechanical process rather than a chemical
one as it involves few molecules at a time. Yet, it is governed up to some
extent by the rules of chemical kinetics and accidentally by inter-molecular
quantum mechanical events 21 .
The rough idea for an experimental identification of an edge in G marked
by a protein P and pointing to a regulatory region R goes as follows: one
replaces the coding part of the gene downstream of R (by means of recombinant techniques described later) with the code for a fluorescent protein with
a short half life. When the gene producing P is activated, then the fluorescence seen in the cell indicates the effect of P on G. Thus, one can detect
enhancer and suppressor edges as well as oriented paths of such edges.
In principle, using microarrays (discussed later), one can detect some
subgraph in G as well as evaluate the differences between such subgraphs
sometimes without knowing each of them.
Post-transcriptional genetic regulation. The actual gene expression, that is
the rate of production of proteins, is not only controlled in the course of transcription but also by the structure of RNA, by its location in the organism,
and a variety of other mechanisms regulating translation. For example, a
particular folding of RNA may slow down the translation process, the chemical environment may influence the rate of degradation of RNA, the usage
of specific codons may effect the rate of translation by ribosomes. Apart
from RNA, a cell may damp gene expression by degrading newly synthesized
proteins.
Replication. The central event in replication is the production of two dou21
A biologically relevant quantum event is the mismatching of complementary pairs
leading to errors in replication. Also, some models of solvents (e.g. water peppered with
small molecules and ions), relevant for binding of macromolecules, are based on semiempirical quantum mechanical rules.
24
ble stranded daughters DNA from a mother double stranded DNA: the two
strands of the mother DNA separates and each of them serves as a template
for a complementary strand, thus each daughter inherits one strand from the
mother, with the other one being newly synthesized. The initial separation
of the mother strands in a bacterium such as E. coli takes place with enzymatic help at a specific initiation site. The synthesis of the new strands
proceeds in two opposite directions starting from the initiation site, where
the moving corners of the resulting growing “bubble” (made by the separated strands) are called replication forks. The synthesis of the new DNA
should go along with the polarization of the template strands: only one of
the strands of the mother DNA agrees with the direction of the movement
of the fork, and templating the other strand is by necessity a discontinuous
process which is divided into the synthesis of relatively short segments in
the direction opposite to the movement of the fork. Systematic jumps of the
double length of the segments allow the process to proceed in the direction
of the fork movement.
The essential role in this process (we omit fine print) is played by DNA
polymerase III, an enzyme built out of 10 subunits with a total mass >
600kDa. DNA polymerase III proceeds with a rate of synthesis of ≈ 1000
nucleotides (≈ 300nm) per second in bacteria, and the full replication process, performed by two polymerase simultaneously (moving in opposite directions), takes about half an hour in E. coli with a circular DNA of 4, 640kb.
The unlinking of the two daughter strands (associated to the two strands
of the mother that are linked in space by the helical winding of DNA) is
achieved with topoisomerase enzymes.
The synthesis of new DNA in human cells proceeds at ≈ 100 bases per
second per fork. The full process takes about 8 hours with the number of
forks estimated between 1,000 and 100,000, where not all forks are active simultaneously during the replication period. The newly born daughters move
to opposite locations in the cell, the cell material is redistributed accordingly
and the membrane undergoes topological modifications eventually leading to
the division of the cell.
There are other replication-like processes modifying DNA besides doubling along with the cell division. For example, some segments may interchange their location within DNA, a (selfish) segment may generate several of
its own copies within the same DNA, viruses may transport a segment from
25
an organism to another, bacteria may exchange fragments of their DNA using
plasmid vectors.
Macromolecular ensembles. Macromolecules in cells and sometimes in vitro
are able to aggregate by self-assembly or by protein aided assembly into intricate geometric and rather rigid structures. Some are suspended in the cytoplasm such as ribosomes, chaperones and proteasomes. The latter are protein
complexes of ≈ 2M Da serving in eukaryotic cells for selective degradation of
proteins. Chaperones are protein complexes (≈ 500kDa) aiding protein folding by disrupting undesirable bonding between different polypeptide chains
as well as between segments of the same chain.
The major global structures in cells are internal and external membranes,
which are made of lipids, polysaccharides and proteins. They control the
biochemical activity of the cell, sometimes actively participating in it, and
also serve to separate different compartments of the cell. Also, they provide
a support for several kinds of molecular motors such as H + driven “turbines”
used in the production of ATP’s and those serving for rotating the flagellae
in some bacteria. Non-membrane based rotatory motors are apparently employed by some viruses (e.g. Bacteriophage Φ29) for packaging their DNA
into a precursor capsid.
Taxonomic structures. The subject matter of taxonomy is constructing various metrics on the spaces of genotypes, species or other biological and biochemical entities (such as proteins and RNAs). There are two different,
mathematically dual, approaches introducing a metric in the (moduli) space
of structured objects. The first is the evolutionary (cladistic) approach, where
the (phyletic) metric is defined by the length of the shortest path of elementary modifications of the object, similarly to the construction of intrinsic
metrics by Gauss-Riemann-Kobayashi. Limiting such a metric to the space
of actually existing biological entities often exhibits a pronounced tree-like
behavior 22 . This is due to the huge size of the space of possible structures,
e.g. of bp sequences of length 109 , where the branches of the evolution process
are very unlikely to come together and form cycles 23 . The second (phenetic)
metric is the phenomenological one, similar in spirit to the Caratheodory
metric on complex spaces. Here, we pick up some distinguished physiological
22
Every graph appears as a quotient of a tree, and the tree-likeness refers to the relative
number of cycles created in the course of factorization.
23
Yet, the tree structure is disrupted by the horizontal transfer of genes.
26
parameters, e.g. size, style of breathing and nutrition, reproduction patterns,
etc., and thus map our space into the space of parameters, where we choose
some simple metric and induce it back to our original space. The dream
of biologists is to independently construct these two metrics such that they
would become equal on the space of the existing organisms.
4
Scales and parameters
One wishes to describe a cell by first identifying and enumerating essential
structures such as DNA, RNA, proteins, cytoskeleton, mitochondria, etc.,
presented in the previous section, and then specify parameters characteristic
for a given cell or a particular state of a cell. When we speak about parameters, we think about points in a rather simple space. Typical parameters are
real numbers taken from a given interval on the line. If we speak of two real
parameters, we want them to be essentially independent and not subject to
a complicated relation. Thus, we admit as a space of parameters a disc or a
square in the plane but not a specific Cantor set like the Sierpinski gasket.
However, if there is an evidence that the values of observable parameters
are restricted to such an intricate set, then this set can be promoted from a
parameter set to a new structure.
The number of relevant (types of) distinct structures in the molecular biology is rather limited, something of the order of 102 −103 , but the dimension
of the parameter space may be very large.
For instance, genomes in a first approximation are words in 4 letters: the
structure is simple and transparent, the space of parameters for this structure
is the space of 4 letter sequences of a certain length. The simplicity of the
structure is due to the fact that we assumed that the individual parameters,
valued at A, C, T, G, may take arbitrary independent values in all positions.
The price we have to pay is the determination of all these letters for each
individual DNA, with no help of additional structure limiting possible values
of the parameters.
In the case of humans, the word has ≈ 3 · 109 letters: even reading
ten letters per second (via chemical analysis) would take about 100 years.
So biochemists learned to read real fast to (almost) determine the human
genome in a mere decade!
There exists an extra structure in the space of (human) genomes: one
27
MACRO- ASSEMBLIES
ATOMS
MOLECULES
MOLECULES
C-C bond
Glucose
1Å
10−10 m
10 Å
10−9 m
1 nm
CELLS
Resolution limit
Red
of light
blood
microscope
cell
Hemoglobin
Ribosome
Bacterium
102 Å
10−8 m
103 Å
10−7 m
104 Å
10−6 m
1 µm
105 Å
10−5 m
Figure 13: Table of dimensions.
chooses a prototypical sequence, a distinguished point in the space, such that
any other sequence differs from it at relatively few essential 24 places. These
variations between humans, probably constitute ≈ 104 − 105 variable letters.
Thus the genome of an individual human, modulo the “universal genome” is
essentially a random sequence of about fifty thousand uncorrelated letters.
Besides the above coding parameters, there is a variety of physical and
chemical parameters characteristic for a functioning cell: size and mass of
geometrically defined structures, characteristic time and energy of localized
interactions, as well as temperature, concentrations of small molecules and
ions, especially acidities (pH level), rates of reactions, etc.
The ranges of these parameters specify different structures of the cell, and
the specific orders of magnitudes of the parameters apparently play a crucial
role in the life of the cell. For example, there are three ranges of energy in
the cell: thermal, weak (e.g. hydrogen bonding) and covalent, roughly of
the order 0.6, 1–5 and 10–100 kcal/mol respectively. See Fig 13, Fig. 14 and
Fig. 15.
If we turn to more sophisticated patterns such as seen in genetic networks,
the distinction between structures and parameters becomes less clear. The
parameters specifying abstract networks are values 0,1 assigned to pairs of
genes depending on whether they directly interact or not. However, this is
24
Here “essential” refers to the biological functioning of the organism. On the other
hand criminologists distinguish different human genomes by variations of nucleotides in
non-essential parts.
28
Noncovalent
bond
ATP
Thermal
0.1
1
Green C-C
light bond
10
Glucose
100
1000
kcal/mol
1
10
100
1000
kJ/mol
Figure 14: Table of energies.
Hinge motion
in proteins
Enzyme-catalyzed
reaction
Unwinding of
DNA helix
10−9
(ns)
10−6
(µs)
Generation of
a bacterium
Synthesis of
a protein
10−3
(ms)
1
103
Figure 15: Time of processes (in seconds).
29
9
not very useful as it leads to a huge parameter space of order 210 . The basic problem is to isolate a structurally simple, admitting a reasonably short
syntactic description, subset in the space of graphs (represented by 0–1 matrices) giving a sufficiently fine approximation to a realistic genetic network or a
class of such networks. In reality, the structure of the network is augmented
by additional combinatorial, numerical and functional ingredients: the space
of parameters is enlarged, yet it helps to identify particular subclasses of
networks actually present in cells. On the combinatorial side, we have a
boolean structure indicating the enhancing/suppressing effect of a gene (or
several genes) on another gene(s). This suggests clustering genes according
to their role in regulation (e.g. position in the network, causality, general
functions, co-regulation, etc.). Also, an individual network is immersed into
a family of networks parametrized by the evolution tree. Browsing through
the tree allows one (by working hard!) to identify persistent patterns corresponding to common functionality of networks across species. Furthermore,
the feasible dynamics of the gene regulation (robustness, polylinearity of kinetics, etc.) imposes constraints on the network which may offset the extra
complexity introduced by dynamic parameters.
The distinction between structures and parameters in biology is blurred
compared to what happens in the physical sciences. A specific set of parameters on one “level of organization” may appear as a governing law at
the “higher level”. (An example in this spirit, is the emergence of symmetry
in the growth of flowers according to Paul Green.) A possible mathematical
formalization of that may appeal to dynamics associated to fractal potentials
with sufficiently separated scales.
5
Design and control of macromolecules in
vitro25
The bacterium E. coli, of volume ≈ 1µm3 , is filled in by cytoplasm, a crowd
of molecules in thermal motion. There are about 300,000 (non ribosomal)
proteins of ≈ 10µm in diameter, 20,000 ribosomes 26 (25% of the cytoplasmic
25
We limit ourselves to polynucleotides and do not touch upon proteins.
The average gaps between ribosomes are ≈ their own size, and the gaps between
(globular) proteins are ≈ twice their size. Proteins constantly hit each other (and run into
26
30
volume), 300,000 tRNA, a couple of thousands mRNA molecules, 50,000,000
small organic molecules including amino-acids, nucleotides, sugars, ATPs,
etc. and various ions. The 70% of the volume is taken by 2 · 1010 water
molecules. (Eukaryotic cells, about 1,000 times bigger in volume, are filled in
by a cytoplasm of similar composition, and they are architecturally organized
by several relatively rigid structures: nucleus, cytoskeletons, Golgi apparatus,
endoplasmic reticulum, vacuole, other organelles, etc.)
In the previous section we gave a rough sketch of what macromolecules do
in the liquid crowd inside a cell, and now we turn to possibilities of what can
be done biochemically in the test tube, or as biologists say, in vitro. Specifically, we want to explain the basic ideas behind bio-chemical manipulations
which allow one to transform information encoded in polynucleotides into
“visible” chemical and physical phenomena.
Imagine the (bio-chemically improbable) situation where we want to distinguish between two species of one-stranded polynucleotides. Both of them
have the same number of bases, say one hundred, and they are represented in
solution in two different tubes. Each tube contains a large (≈ 1018 ) number
of copies of the same molecule, and we want to decide which tube contains
which sequence. The macroscopic properties of the two solutions are essentially indistinguishable since overall chemical and physical properties of two
polynucleotides of equal length are very close to each other (at least if the sequences are not too special). However, if we recall the Crick-Watson duality,
we come up with the following obvious solution: prepare a third tube with
the double amount of polynucleotide molecules complementary to those in
the first one, and pour half of its content in the first tube and half into the
second. The complementary polynucleotides in the first tube will hybridize
forming double stranded molecules, while in the second tube they will remain single stranded. Since the physical properties of the molecules in the
two tubes became sharply distinct, this can easily be seen in the behavior of
a variety of macroscopic parameters of the two solutions: viscosity, optical
properties, specific heat capacity, resistance to degradation by single-strand
specific nucleases, etc.
To go further, besides the Crick-Watson complementarity controlling the
hydrogen bonds, we need to break and create covalent bonds. In the cell,
this is done by a variety of enzymes. Some of them are “universal” and do
ribosomes) with small molecules snicking in the gaps between them.
31
1.
2.
Figure 16: A nicked double stranded DNA (top), and the result of the action
of ligase (bottom).
not discriminate between specific bases, and some others are site specific, i.e.
their action takes place where a particular short subword is present. Here
are a few examples:
Ligase: given a double stranded DNA with a “missing” covalent bond in
one of the strand, this enzyme “constructs” the bond. Besides nicks, it also
repairs double stranded breaks, albeit less efficiently. See Fig. 16.
DNA polymerase: Consider a one-stranded DNA together with a short complementary word, called a primer, coupled to the “left” end of the DNA
(for properly understood left/right on oriented DNA). Suppose that there
is a supply of free nucleotides in the solution. Then, the DNA polymerase
constructs the full complementary chain as illustrated in Fig. 17.
Topoisomerase II: when two segments of double stranded DNA come close
together, the topoisomerase cuts the covalent bonds in one of the segments
and then join them again on the other side of the second segment as shown
in Fig. 18.
Recombinase: see Fig. 18.
Restriction enzyme EcoRI: it recognizes a double stranded DNA at the specific site GAATTC and cuts it as indicated in Fig. 19. Such a cleavage leaves
free two short complementary single stranded segments, called sticky ends.
Several cleaved DNA’s can recombine by cross hybridization of the sticky
ends. This step, followed by ligation, produces new DNA’s.
32
TAGGCAT
1.
ATCCGTAAGTGGAACCTGATCGTACGTGGAACACACATG
A
A
2.
N.
C
TAGGCATT C
ATCCGTAAGTGGAACCTGATCGTACGTGGAACACACATG
TAGGCAT TCACCTTGGACTAGCATGCACCTTGTGTGTAC
ATCCGTAAGTGGAACCTGATCGTACGTGGAACACACATG
Figure 17: DNA Polymerase.
Figure 18: On the left, the action of topoisomerase II and on the right, the
action of recombinase. The double arrow indicates that the operations are
reversible.
cleavage
1.
GAATTC
CTTAAG
sticky ends
2.
G
AATTC
CTTAA
G
Figure 19: A double stranded DNA before and after the cleavage by EcoRI.
33
DNAse I: it cuts double stranded DNA at sites specified by a certain class
of subwords. The words of this class are characterized by physical properties
of the corresponding stretches of DNA, especially by their flexibility. A
precise determination of this class remains unknown, largely because of the
complexity of the involved bio-mechanical problem.
The above enzymes can be used in vitro and allow the following manipulation with DNA strands.
Synthesis of polynucleotides. In order to chemically realize a given sequence, say TGCAATTCG, we start with T attached to a solid support by
one of its ends and add a modified G to the solution so that G can covalently
bind to the free end of T, but no G can bind to G. After TG is formed, the
unattached G’s are washed out from the solution, the “blunt” end of G is
converted to its active form, modified C’s are added to the solution, and so
on. In order to have a large area of the solid support, one uses as a support
microscopic beads suspended in the solution, or a microporous glass, with
pores ≈ 1µm and area 1018 nm2 per 1cm3 of volume, allowing in practice up
to 1019 growing molecules on the surface.
One can build up oligonucleotides of 100-200 nucleotides in length. At
every stage of the process, some oligo might not acquire the added base and
the synthesis protocol includes a step, called capping, in which unreacted
growing ends are “chemically disactivated” to prevent further growth. This
prevents the appearance of long erroneous oligos (except for errors occurring
at the end of the synthesis) and makes purification easier. Yet, the sequential
accumulation of errors makes a reliable synthesis of longer oligos impossible
by this process.
Polymerase Chain Reaction (PCR). This is in the league with the invention of the wheel and the nuclear chain reaction.
Given several molecules of a double stranded DNA in a tube, one can
amplify their number exponentially in time along the following lines. Add
to the tube N (of order 1019 ) copies of two different single stranded oligonucleotides of the length about 10 bases, identical to the starting words of the
two strands of DNA. Also add a generous amount of nucleotides and DNA
polymerase. If we raise the temperature to ≈ 100◦ C, the DNA denaturates,
i.e. its strands fall apart. When we sufficiently lower the temperature, our oligos will hybridize to the corresponding complementary ends of DNA strands
and will play the role of primers (this process will be in thermodynamical
34
primers
ATTCGC
CGTTAA
TAAGCG
CGTTAA
ATTCGC
GCAATT
TAAGCG
CGTTAA
1.
ATTCGC
2.
CGTTAA
ATTCGC
GCAATT
Figure 20: A double stranded DNA and the oligos corresponding to the beginning ends of its complementary strands. After the strands are separated,
the oligos hybridize to the ends.
competition with the hybridization of the two strands among themselves,
but since we have an excess of oligos, most DNA strands will hybridyze with
primers). (See Fig. 20) Then the DNA polymerase synthesizes a complementary strand to each of our original strands and thus the total number of
strands doubles. This works reliably even after many iterations for relatively
short DNA’s, several hundreds bps long, and there are various modifications
of this method allowing reliable amplifications of DNA up to 30,000 bps long.
The amount of DNA obtained is limited by the number of primers (and of
free nucleotides, of course).
The presence of other DNA’s, different from the ones beginning with a
given word, represented by the oligos does not essentially affect the method:
only the relevant DNAs will be amplified. This allows in particular to tell if
a single molecule of a particular DNA is present in the tube among zillions
of other DNAs. This is a useful test in biology as well as in criminology.
Sequencing. We start with a sample of one-stranded DNA (which can
be amplified as needed using PCR) where we assume for simplicity of the
exposition that this DNA is coupled with a primer at the beginning end. We
add DNA polymerase, an excess of bases A, T, C, G into the tube and a small
35
amount of A , that is a chemically modified A with the following property:
if A terminates a polynucleotide chain, then no new base can be attached to
A and the synthesis stops at A . After polymerization, we obtain a variety of
one-stranded DNA segments, complementary to initial subwords of the original DNA strands, where each segment terminates at A . We separate the
new strands from the old ones by raising the temperature and determine the
spectrum of their masses or lengths by some physical/chemical method. For
example, we can ionize our molecules (with reasonably controlled charges)
and accelerate them in vacuum by an electric field, and follow their trajectories in a magnetic field. These are distributed according to mass and we can
determine the masses of our segments with high precision. Alternatively, we
can make the molecules going through a gel where their speed is inversely
proportional to their length, and thus determine the length spectrum also
with high precision. In either case, we determine the length of the segments
terminating with A which correspond to the positions of T ’s in the original
DNA strand. This procedure applies to the remaining letters T, C, G and
provides the position of all four bases in DNA.
Molecular beacons. Suppose we have a solution of various one stranded
DNA (more realistically, RNA) in a tube and we want to decide whether a
particular short word, say of 15 nucleotides, appears as a subword in some
of them. One prepares a large amount of oligo of 25 bases which contains in
the middle a complementary 15 letters word pinched between two mutually
reverse complementary segments, each of 5 letters long. At room temperature, such molecules make hairpins with circular loops of 15 bases long and
double stranded stems of 5 base pairs. (Compare to the picture in the right
hand side of Fig. 7). If we add this oligo to the solution, each loop which
finds the complementary word will hybridize with it, forming a rather rigid
double stranded DNA. As a result, every such loop straightens up and the
two ends of the oligo separate.
The separation of the ends is detected by chemically attaching a fluorophore to one end of the oligo and a quencher to the other end. When
exposed to ultraviolet light, the fluorophore emits visible light unless it is
close to the quencher which absorbs this light (and turns it into heat). Thus
luminescence appears if the subword is present in DNA and some of the loops
are open.
The oligo with the fluorophore and quencher attached to its ends serves
36
as a molecular beacon signalling the presence of our word. This device is
used, in particular, as a diagnostic tool for the detection of mutations in
DNA (RNA).
Microarrays. These serve to determine the level of expression of N (possibly all) genes in a given cell culture or tissue by measuring the concentration
of the corresponding mRNAs. Here is the idea. Take an array of N small
tubes and put into the i-th tube a solution of molecular beacons complementary to a segment in the i-th mRNA. Add to each tube the solution of
mRNA extracted from cells. Then, the intensity of light in the i-th tube
will be (roughly, but not quite) proportional to the concentration of the i-th
mRNA in the solution. In practice, one does not use molecular beacons (yet)
but rather attaches fluorophore to the mRNA extracted from the culture,
and measures fluorescence resulting from hybridization by different means
(without the use of quenchers). To increase reliability and compensate for
errors (partly arising from the drive for micro-miniaturization), one often
compares expressions from two different cultures, marked by different fluorophores and mixed together. This mixture is added in each small tube and
the color of the fluorescence witnesses the ratio of the expression.
For basic research, the main purpose of microarrays is to reconstruct the
gene networks in a cell, but this goal faces many unsolved problems. The
present microarrays are not sufficiently reliable, due to a variety of uncontrolled errors: self-hybridization of RNA affecting the hybridization with
the probe, dependence of hybridization on the fluorescence dye, difficulty in
synchronizing cell cycles and/or making time dependent measurements (the
latter problem does not exist for molecular beacons), presence of dust particles and/or spontaneous fluorescence, etc. The major problem is that, due to
post-transcriptional regulation, the production of a particular protein is not
proportional to that of the corresponding mRNA and thus mRNA microarrays should be eventually complemented by protein-chips, where a protein
is detected by binding to its antibodies, i.e. a specifically designed protein
to which a given protein binds. Even granted that all these technical problems are resolved, the full combinatorial problem of reconstructing the gene
network from the averaged (!) expression levels, under variable conditions,
remains wide open 27 .
27
These difficulties do not play a significant role in certain practical applications of
microarrays such as fast genotyping, where direct measurement might be sufficient, and
37
Recombinant techniques. One can use the bacterial replication machinery
for cloning long strands of DNA, which allows a higher fidelity than PCR.
This machinery can be also used for generating and selecting sequences coding for proteins with desired enzymatic and/or binding properties, e.g. antibodies for specific pathogens.
Given a segment of double stranded DNA, up to several tens of thousands
base pairs long, it can be inserted into the genome of a (bacterial) cell by
a variety of recombinant techniques, also called genetic engineering. This is
done with either plasmids28 or viruses29 used as vectors carrying the desired
DNA segment into the cells. The vector DNA is cleaved with a restriction
enzyme and ligated to the DNA segment. (The latter is prepared with complementary sticky ends which allow the hybridization of the segment to the
vector and the following ligation.) Then, the resulting recombinant vector is
inserted into the cell. This is done either with the help of a chemical making
the membrane permeable to plasmids, or by shooting particles coated with
plasmids into the cell culture. A viral vector penetrates into the cells by itself
using its coating proteins.
A common technique for selecting the cells which did receive recombinant
vectors, consists in inserting an auxiliary gene into plasmids, where the gene
makes the receiving bacterium resistant to a particular antibiotic. When the
treated cells are raised up on a nutrient containing the antibiotic, only the
transformed cells will survive. In protein engineering, one “marks” transformed viruses by inserting the code of a specific protein next to the gene
of a coating protein. In the course of translation, the new protein will fuse
with the coating protein and will be displayed on the coat of the virus. Then
one prepares a “complementary” protein with a special affinity to the fused
protein and attaches it to a solid support. When the support is brought to
in some diagnostic procedures, where a rough gene clustering suffices.
28
Plasmids are circular double stranded DNA molecules disjoint from the chromosomal
DNA in cells, and vary from a few thousands to more than 100 thousands bps in length.
They use the translation machinery of the (host) cell and replicate along with the cell
division. They occur naturally in bacteria and yeast, for example.
29
A virus particle, called a virion, is a self-assembling macromolecular complex constituted of one or several polynucleotide molecules covered by a protein coat. Viruses
penetrate into prokaryotic or eukaryotic cells, throw away their coats and use the cell
machinery to replicate and to produce viral proteins (the number of proteins encoded in a
virus ranges from as few as 4 up to roughly 200). Some viruses live in the cytoplasm and
some incorporate their DNA into the chromosomal DNA of the host.
38
contact with the viral culture, the marked viruses bind by affinity and are
extracted from the solution.
Apologies. The above description of biochemical techniques as well as our
overview of the cell functions should not leave an expression of being anything
close to exhaustive or in the front-line of research. There is a body of classical techniques such as NMR, x-ray cristallography, relaxation spectrography,
emission spectroscopy, and various microscopy techniques: electron, cryoelectron, scanning transmission electron, scanning tunneling, atomic force,
etc. We did not touch upon proteomics, microscopy for subcellular localization of fluorescent markers, design for unicellular studies, single molecules
experiments, and many many other directions.
6
Formalizable structures
One seeks for a coherent 30 network of models, i.e. consistent mathematical theories, reflecting the behavior of macromolecules and of ensembles of
those in vivo and in vitro. A biologist hardly expects models comparable in
abstractness and universality to those in physics (such as statistical thermodynamics, spectral theory of Schrödinger equations, etc.) but rather tries to
isolate regular macroscopic and mesoscopic patterns in biological systems in
order to predict and design experiments.
Let us start with a list of patterns which invite a mathematical framework:
Repeativeness and partial symmetry of stochastic motives. The very existence of biology (and any rational science in general) depends on the systematic repetition of patterns, motifs and structures, which allows compression
of the observed information. Given a large collection of biological macromolecules, macromolecular complexes, cells, organisms etc., one can divide
them into relatively few groups with great similarities between members of
the same groups. In fact, this phenomenon starts from biochemistry, where
the number of small molecules in a cell by far exceeds the number of different species of molecules (i.e. amino-acids, nucleotides, sugars, etc.), and
where large molecules (i.e. polynucleotides, proteins and polysaccharides)
30
The inter-connections between theories do not have to be fully formalized. Sometimes,
one does not even expect that they are formalizable at all, as for the “quantum ←→
classical” correspondence, for instance.
39
display similar repetitiveness of basic motives. Besides, even literally different bio-molecular aggregates, e.g. cells, share many common features in
their design and functionality. This suggests that the same low variability of
combinatorial and dynamical patterns will emerge in other structures, e.g.
gene networks, once these are revealed.
The symmetry in biology is of a different nature than that in physics,
where one sees statistically completely homogeneous media such as gases and
liquids as well as truly homogeneous symmetric structures in crystals. In biology, stochastic patterns, e.g. organisms of a given species, appear faithful
to fine details by far more often than one might naively expect on the basis of
physics and probability theory. Yet, this phenomenon should be eventually
explained in terms of locality and stochasticity. More specifically, why accidental low free energy metastable states harbor stochastic symmetry? How
the major biological sources of symmetry, which are templating, replication,
and universality of production, can be derived from general energy/entropy
considerations?
Channeled relaxation. In many situations a macromolecular system minimizes a complicated function of its parameters in an amazingly short time:
RNA and proteins fold in several minutes 31 , macromolecular complexes selfassemble in cells, and sometimes in vitro, almost as fast, covalent bonds are
destroyed and created in a fraction of a second in the presence of enzymes,
the reproductive efficiency of bacteria has grown from 0 to 2 divisions per
hour in mere two billion years! Eventually one wants to understand this phenomenon as a feature of the function being minimized, or rather of families
of functions.
31
This should be confronted with the rate of individual molecular events: the hinge
motion of proteins happens on the scale of nanoseconds while molecular vibrations are
thousands, sometimes millions, time faster. However, these speeds cannot significantly
shorten the rate of folding of long polypeptides since the number of possible configurations
grows exponentially with the length: to explore 2N configurations of chains of length N
with a rate of 1015 moves per sec, one needs 2N −50 seconds. On the other hand, the
number of deep and wide local minima may grow significantly slower than 2N , thus allowing
fast folding by the gradient directed random walk. An argument due to Eigen (private
communication) and Kauffman suggests that adding new dimensions to the configuration
space, predominantly moves local minima (of the energy function) to saddle points and
thus a random function in many variables may have fewer “dangerous” local minima than
a naı̈ve counting suggests. A comprehensive mathematical development of this idea is still
missing.
40
Complementarity and self-assembly. Both transcription and DNA replication are based on templating by the Crick-Watson complementarity. Also,
binding of proteins (e.g. between a protein and its antibody) can be seen as
a complementarity pairing between macromolecules.
Mathematically speaking, there are (partial) involutive symmetries in the
macromolecular world, which bring to one’s mind reflection groups. For example, the coats of viruses with icosahedral symmetry, self-assemble this way,
out of several identical protein molecules 32 . Less symmetrically, ribosomes
self-assemble, even in vitro, if the RNAs and proteins constituting them are
present in the solution.
It happens sometimes, e.g. for viruses, that the process of self-assembly
goes in stages, where the initial assembled structures serve as a scaffolding
for the following one and then disappear from the final result.
A simple model for self-assembly is provided by a collection of convex
(flexible) polyhedra in the space, with prescribed affinity between some of
their faces. Thrown into solution, these polyhedra may (or may not) assemble
into larger structures similar to covering of orbihedra.
Compartmentalisation. A cell, especially a eukaryotic one, is far from being chemically homogeneous. It is divided into semi-connected compartments
and it is filled with filaments channelling and enhancing the biochemistry of
the cell. Both the walls of the compartments and the filament channels can
be static, kept together by chemical bonds, or virtual, supported by the dynamics of the cell. Compartmentalisation inhibits the global relaxation of the
system but enhances partial relaxations within the compartments. The cell
itself is defined as a compartment formed by a semi-permeable membrane,
decoupling the dynamics within and without the cell.
One wishes to see this picture in a high dimensional (time-)space as a
kind of mediating topology depending on auxiliary parameters such as the
permeability of a given substrate through the membrane.
Homeostasis and replicative stability. Homeostasis, a chemio-dynamical
stability of a cell in the variable environment, is disrupted by replication.
The cell is brought out of a stable regime, dynamically speaking an attractor, but the stability is regained in a different framework: the dynamical
32
Chirality of biomolecules does not allow (orientation reversing) reflections, but rotation
groups do appear as symmetries of crystallized proteins and of coats of some viruses, for
example.
41
features of the system reappear in the multiplicity of nearly identical cells
represented by the cartesian product of many copies of the attractor, and
ensure preservation of information by perpetuation. In fact, homeostasis of
an individual cell cannot be stable for a long time 33 as it would be destroyed
by random fluctuations within and without the cell. There is no adequate
mathematical formalism to express the intuitively clear idea of replicative
stability of dynamical systems. (A possible model might use dynamical time
of variable fractal dimension depending on the number of bacteria in the
population, expressing the idea of weak correlations between time-clocks of
different bacteria.) It is unclear if the appearance of many copies of similar
individuals is unavoidable or just a transitory feature of life. The question
is whether there exists, mathematically speaking, a stable highly organized
system with no high repetition of structural components, similar in spirit to
Oceanus sapientissimus of Stanislav Lem.
Information and amplification. Among other biochemical parameters of
the cell, the information encoded in DNA is distinguished by the following
features:
- it is time conserved 34 . More precisely, it changes with much slower rate
than other parameters;
- unlike most quantities conserved by physical systems (e.g. energy, momentum, etc.), the “information observable” ranges in a very large space: the
space of 4 letters sequences of significant length;
- small perturbations of information unpredictably amplify. The dynamics of
the cell maps small “information sets” onto much larger spaces of biochemical
and morphological parameters. Besides expanding the size, this map may
dramatically increase the structural complexity of the information set.
What is a proper mathematical description(s) of the paradigm “genotype determines phenotype”? Would every consistent mathematical model
require a notion of “inheritable initial condition” corresponding to what biologists call “epigenotype”? Can the above properties be turned into precise
definitions? If there is a rigorous model, are the three properties mutually
33
In a protected environment, a cell, such as a neuron in the human brain, may live for
a 100 years.
34
An essential property of information which is commonly emphasized, is its syntactic invariance, that is the relative independence of the specific physical carrier. Such a
property can be hardly formalized in a dynamical framework.
42
independent in it? Is the presence of information/memory unavoidable in
any life-like dynamics?
Universality of production: translation and replication. The two basic
cell machineries, DNA replication and synthesis of proteins, are reminiscent
of the universal Turing machine. DNA replication works on any given piece
of DNA (viruses know it only too well). The same almost applies to transcription and translation: DNA is roughly divided in two parts. One encodes
house-keeping genes coding for proteins supporting the basic machinery, e.g.
ribosomal proteins. This house-keeping DNA cannot be arbitrarily modified without badly disrupting the cell functions. On the contrary, one can
change the remaining part of DNA by inserting and deleting DNA fragments,
thus making the cell to synthesize any given protein (unless it happens to be
poisonous to the cell).
Can this universality and interchangeability be expressed in general mathematical terms, e.g. in the language of dynamical systems as a pseudo-group
(or something like a category) of partial symmetries?
Separation of energy scales. The functioning of the cell can be thought
of as a continuous flow and redistribution of energy which comes in three essentially different forms: thermal energy, (weak) binding energy, and covalent
energy.
The thermal energy is a random field in space, where (not very) large
fluctuations are comparable to the binding energy but much weaker than the
potential barriers encompassing the covalent energy. If not for the mediating
effects of binding energies, there would be no significant thermal dissipation
of the covalent energy (at the room temperature) on the time-scale of the cell
cycle 35 . The system would behave as if it were at equilibrium. A familiar
example is a water solution of H2 O2 which is rather stable at the room
temperature and away from light; enzymes from saliva act as catalysers and
make H2 O2 to decay with a release of free oxygen visible in the bubbles if
one spits into the solution.
For us, the covalent energy is a scalar distribution of labelled points in
space, where the geometric polarization of the bonds is ignored and where
each label corresponds to a species of small molecules. The label should
specify the covalent energy of a molecule as well as the activation energy,
35
Some covalent bonds can be spontaneously broken by hydrolysis with a release of
thermal energy, but we do not consider this at the moment.
43
that is the height of the potential wall preventing the release of the stored
energy 36 . The relevant information is encoded in density functions of labeled
points in the 3-space.
The binding energy applies to interaction between macromolecules and
other large and small molecules. (The weak interactions between small
molecules are ignored here, or delegated to the thermal energy.) Unlike the
thermal and the covalent energy, the binding interaction on the nanoscale is
seen not as a scalar but rather as a vector depending on the mutual position
of the molecules.
The system governed only by thermal and binding energies (without gain
or loss of covalent energy) would relax to the thermal (quasi-)equilibrium
relatively fast compared to the cell cycle 37 . Deadly on the scale of the cell, the
“rigor mortis” following relaxation, is incorporated into the local dynamics
for assembling skeletal structures with specific geometries depending on the
polarization of binding energies.
The structure produced by the binding energy channels the distribution of
the covalent energy and facilitates its exchange (metabolic pathways) and release. The released covalent energy is harnessed to build up macromolecules.
Some part of the covalent energy is redistributed among (macro)molecules,
some is converted to the binding energies of the macromolecules, and the rest
dissipates to heat. The changes in binding energies bring in dynamics to the
geometric structures created by them.
It seems that no organized structure can exist in the presence of only
two levels of energy. The sizes of the gaps seem also important: it is hard
to imagine self-organizing structures employing nuclear energies 38 instead of
covalent ones, but there is no mathematical model where this would become
36
This is an oversimplification: some covalent energy can be stored in macromolecules.
Besides, it may “reside” in pairs of molecules as in the gaseous mixture of H2 and O2 for
example.
37
Death by asphyxiation (starvation) may be postponed by sporulation.
38
The sources of nuclear energy, external to biological systems, such as the sun and the
inside of the earth, are crucial for sustaining life. These would be deadly in the pure form
of high energy photons and they become acceptable only after intermediate transformation
to the thermal energy corresponding to green light photons and below. The problem is
whether the nuclear energy can be structurally incorporated into a life-like system where a
high level of energy is tempered by sparse spatial distribution but without transformation
to lower level thermal energy. The obvious intuitive answer is negative and it would be
nice to have a general mathematical theorem confirming the intuition.
44
a theorem.
Specificity of scales and numbers. To demonstrate the importance of specific relative values of numerical parameters in the functioning cell (numbers
of particles, their size, mass, energies and interaction/relaxation time scales)
we shall draw a conclusion from the following simple observation: there are
as many microns (a bacterium size) in a centimeter (a small tube size), as
angstroms (an atom size) in a micron.
“Theorem”: Consider a random pond of water of 1000m3 containing the
chemical compounds needed for life and an excess of free energy. Then, with
2
probability P ≥ 1 − 10−10 , it contains life in unicellular form.
The proof depends on a “lemma” and an extra assumption.
“Lemma”: There exists a bacterium B ≤ 1µm3 in volume where the probability of replication within a unit time interval ∆t is 10 times greater than
the probability of death within ∆t .
To rigorously prove the lemma one needs a formal description of a bacterium B, e.g. E. coli. We do know that E. coli exists 39 and the biologists
are currently preoccupied with furnishing such a description.
“Simplifying assumption”: When a bacterium divides, its daughters are
separated in the pond and do not interact with each other except for sharing
common nutrients. In particular, one bacterium cannot kill another one,
even allowing mutation and evolution.
This assumption does not correspond to the evolution of unicellular organisms on earth: one of the multicellular descendents of the primordial cell
is nearly able to wipe out the bacterial population inhabiting the 1018 m3
“pond” of the biosphere.
“Proof ”: Since 1µm3 contains ≤ 1012 atoms (and small molecules such
as H2 O, CO2 , N2 and with luck NH3 ), the probability of a spontaneous
39
E. coli bacteria have volume 0.6 − 3µm3 and they are not able to sustain their existence without the support of photosynthesizing bacteria. The latter, the present day
cyanobacteria, are rather large, ≈ 100µm3 . On the other hand, there are other bacteria,
called mycoplasma, that can be as small as 0.02µm3 . But these usually live parasitic existence in association with animal and plant cells. Cells of the size of mycoplasma and with
metabolic abilities of cyanobacteria are ideal for our lemma. Such cells are believed to be
among the first inhabitants on earth.
45
12
assembly of B out of atoms, within ∆t , e.g. one hour, is 10−12·10 , since
each atom out of 1012 can occupy 1012 positions in space. On the other
hand, the pond can be happily occupied by 1016 bacteria B, where each
1µm3 bacterium is allowed 105 µm3 living space.
Since 1/2 1/10, once B appears, its descendents populate the pond
with probability P ≥ 1/2. Also, the probability of death of the full popula16
16
tion of bacteria in the pond is ≤ 10−10 . Hence, Plif e /Pdeath ≥ 1/2 · 1010 ·
12
10−12·10 . Q.E.D.
Remarks. If we scale the size of organisms linearly by k, the required
living space grows as k 2 . For example, to make the above work for a 1cm
size organism, one would need a volume of ≈ 1020 km3 which by far exceeds
the volume of earth.
The above “theorem” does not say that life appears from non-life with
high probability in a short stretch of time but rather that the average number of “alive states” (i.e. states where the system contains a living cell)
in the configuration space exceeds that of the “dead states”. The paradox
appears only if one misuses the ergodic theorem and interprets the time average as something observed in realistic time intervals. In our example, huge
time intervals where the system has no life will interchange with even larger
intervals 40 with life. The boundary of a long interval is negligibly small compared to the length of the interval, making the transition probability very
low. But since the action takes place in a high dimensional configuration
space, the boundaries of relevant regions (i.e. regions of alive states) become
much larger relatively to the volume of these regions (isoperimetric concentration phenomenon). Intuitively, the large dimension allows many scenarios
for non-life/life transition. See the end of Section 7 for a mathematical discussion.
The probability of a configuration of atoms depends on the energy via
the Gibbs factor. This has no significant effect on our computation but the
presence of potential barriers (activation energies) aggravates the transition
problem from “dead” to “alive” states.
40
Maxim Kontsevich pointed out to us that the earth is likely to turn into a blackhole
within such a time interval: we must exclude gravitation from the model. This is a mild
restriction compared to an unlimited supply of suns, stability of proton, etc. that are
needed for our “proof”.
46
The probability of an “alive” configuration is by far greater than 10−N for
N = 1013 , since there is more than one “alive” configuration: the potential
number of possible self-replicating configurations is something like 10α·N ,
where α is a small, but not negligibly small, positive number. This α roughly
represents the percentage of rearrangements of atoms keeping an “alive”
system alive: out of N atoms, α · N of them can be independently modified.
The number α can be interpreted as a relative dimension of “life” and
its realistic evaluation would be quite interesting. The idea of dimension
can be seen in a M × M regular square array of points in the plane, where
1
M = 10 2 N . Subsets containing 10α·N points for α = 12 corresponds to lines
in the square and have relative dimension 12 .
Self-replication can be thought of as a fixed point of a certain transformation (dynamical system) modelling replication. A certain set of replicative
systems make an attraction basin to the set of (stably) self-replicative systems, and the relative measure of this basin maybe reasonably large to allow
a time realistic scenario for transition from non-life to self-replicating life.
Replication may represent, for example, a division of a membrane-bounded
entity (e.g. a micelle or a liposome) into daughters far from being identical
to the mother and to each other, and in the course of sequential divisions,
they structurally converge to self-replicating organisms.
All ten features mutually intertwine and our presentation of them will be
non-linear.
7
Models and problems
Populations of strands and hybridization diagrams.
An AG-strand, or just a strand, is a directed finite linear graph with
letters A, A−1 , G, G−1 labelling each edge. This is either an oriented segment
or a cycle (a topological circle) with letters written on it. In other words, it
is either a path or a cycle 41 in the figure ∞ representing the free group on
two generators.
41
A path in a graph (e.g. the figure ∞) is a combinatorial map of the subdivided
interval into the graph, and a “cycle” refers to a map of a subdivided circle. These maps
are allowed to fold, i.e. they are not assumed to be locally one-to-one. In the figure ∞,
this corresponds to the possibility of reducible words such as AGG−1 A.
47
Figure 21: Surfaces associated to a strand (left) and to a circular strand
(right).
A population of strands is a measure on the set of strands. Warning:
as we want to model populations of strands in realistic solutions in tubes,
where the measure weight attached to a strand equals the number of the
copies of the strand in the solution, one must take into account the fact that
the number of strands in a population (1020 − 1030 ) is large compared to
the number of all possible short strands (30 − 40 letters long) but abysmally
small compared to the number of strands of length more than 100 letters. A
better model for a population of long strands is a random measure on the
space of strands.
combinatorial
A hybridization diagram D is a graph D together with a map of a disjoint union of strands onto it, denoted h :
i∈I Si → D,
where the Si are strands indexed by I, and where the following conditions
are satisfied: vertices go to vertices and edges onto edges, thus every strand
Si is realized by a path or by a cycle in D. Each edge in D has as a pullback
either a single edge or two edges with opposite orientation and reciprocal
labels. The latter is called double stranded edge and the former, is called
single stranded. 42
Two edges in D are strand connected if they both descend from the same
strand Si by the map h. A hybridization diagram is connected if every edge
can be reached from another one by a chain of mutually strand connected
segments. (See Fig. 26.) Thus, every diagram divides into connected components which can be a priori smaller than the topological components of D.
We specifically assume that the underlying topology of D actually defines
the same components as the strand connectedness.
Hybridization (involution). Two edges in the union i∈I Si are hybridized
42
We do not distinguish D and D insofar as it does not lead to confusion.
48
Figure 22: Two examples of hybridization surfaces.
if they have equal images under h in D. Hybridized edges
come in reciprocal
pairs with a natural involution h̃ on the union Shy ⊂ i∈I Si of those
pairs.
The graph D can be described in these terms as the quotient of i∈I Si by
the involution h̃.
Hybridization surfaces. Given a strand S, we consider the surface CS ,
that is defined as S × [0, 1] for circular strands, and for linear strands as
S × [0, 1] with the two edges corresponding to the end points of S shrunk to
single points. (See Fig. 21.)
The hybridization surface CD associated to the diagram D = (D, h :
i∈I Si → D), for Si = Si × 1, is defined by attaching the disjoint union
of the surfaces CSi to D via the map h. This is indeed a topological surface containing D and naturally retracting to it. The boundary of CD is
subdivided in two parts: one corresponds
to the non-hybridized part of the
strands Si ’s, and the second part, i.e. i∈I Si × 0, decomposes into segments
with disjoint interiors corresponding to the Si × 0. (See Fig. 22.) The labelling map i∈I Si → ∞ naturally extends to a continuous map CD →
∞ . Conversely, a hybridization diagram can be defined as a surface with
a distinguished oriented part of the boundary and a labelling map of this
surface to ∞ .
The population of diagrams is a measure on the set of connected hybridization diagrams. It represents the numbers of species (i.e. isomorphism
classes) of macromolecular hybridization aggregates in a solution. Here, even
49
more than for strand populations, one must be careful with the stochastic
interpretation of this measure. A true stochastic object is a random population of diagrams, where the weights assigned to the diagrams are not real
numbers but positive random variables, or better, it is a probability measure
on the space of populations of diagrams.
Dynamics of hybridization. Dynamics of hybridization refers to a random
walk in the space of diagrams, reflecting what happens to an ensemble of
strands in a solution. At every step of the walk, a diagram transforms to
another diagram with a certain probability weight assigned to the step, where
each transformation is a hybridization or a dissociation of an edge. A full
description of such a walk should take into account the interaction energy
between different letters, corresponding to actual energies of hydrogen bonds,
the bending energy, etc.
The stationary states of the corresponding Markov chains can be modeled by a Gibbs measure (defined later in the section), with proper entropic
weights assigned to these states. The first question (corresponding to the
zero-energy) is finding the minimum energy diagrams. Let us simplify further, ignore the bending energy and prescribe the (local) energy at every edge
as follows: every non-hybridized edge has energy zero; to every hybridized
AA−1 pair is assigned a certain negative energy e(A), and to every hybridized
GG−1 pair is assigned a negative energy e(G). Besides, one can partly take
into account the bending by assigning certain positive energies to the vertices
of the graph. For example, if the hybridization map folds (i.e. it is not locally
one-to-one) over some vertex in D, then one assigns infinite positive energy
to this vertex to exclude such a folding.
The total combinatorial energy of the diagram is the sum of energies of
all edges and vertices. In some cases, the minimum energy diagram can be
easily found. Here are some examples.
Complementary strands. If S and S −1 are complementary strands, then
obviously the minimum of the energy is achieved by the linear or the circular
diagram hybridizing S with S −1 . This minimal energy diagram is unique if
we agree to assign (arbitrarily small) positive energies to the vertices, except
for circular strands where the corresponding word has a non-trivial cyclic
symmetry, and where the number of non-isomorphic diagrams equals the
order of the (cyclic) symmetry group of our word with the two ends identified.
(See Fig. 23.)
50
Figure 23: Circular hybridization of two non-circular strands containing nontrivial symmetry in their words.
Y
Y -1
X
X -1
Z
Z -1
Figure 24: Minimal energy hybridization of the three words XY , Z −1 X −1
and Y −1 Z.
51
Triple junctions. Consider three sufficiently long random words X, Y, Z,
and take the three strands XY , Y −1 Z and Z −1 X −1 . The minimal energy
diagram is unique and it is depicted in Fig. 24.
These two examples indicate how the presence of symmetry and of randomness may effect the hybridization behavior of polynucleotides. A general question is the characterization of those ensembles of strands where the
minimum energy diagram(s) can be explicitly described. Another question
concerns the hybridization of diagrams Di where the Crick-Watson pairing
applies to the remaining single stranded edges of the diagrams. In other
words, we consider the space D({Di }) of the diagrams which can be obtained by the hybridization of the Di ’s. This space naturally embeds into
the space D({Si,j }), where Si,j are the strands making Di . In this notation, the problem consists in minimizing the free energy on the subspace
D({Di }) ⊂ D({Si,j }); thus, the minimal energy of the diagram hybridization
is typically greater than that for the strands making the diagrams. Examples
of specified multi-stage hybridization systematically appear in the chemistry
of self-assembling DNA nano-devices and DNA computational schemes.
The moduli space of diagrams. Given a collection S = {Si } of strands,
we define the moduli space D = D(S) as a directed graph whose vertices
are diagrams D made of Si ’s and whose edges D1 → D2 correspond to
hybridization: the diagram D2 is obtained from D1 by gluing together a
pair of free edges. The moduli space D has a distinguished source vertex
corresponding to the non-hybridized strands, and many sinks corresponding
to fully hybridized strands.
The graph D(S) is almost (modulo possible automorphisms of S) entirely determined by the four numbers NA , NA−1 , NG , NG−1 of the occurrences of the four labels in S, since the Crick-Watson pairing is irrespectful
of the topology of S and of the position of the labels. As a mere graph,
D(S) = D(NA , NA−1 , NG , NG−1 ) looks rather transparent but it comes along
with additional (not so transparent) structures. In fact, D serves as the base
space of a “fibration” f : D̃ → D where the fiber f −1 (D) ∈ D̃ is just the
graph D underlying D, and where the f −1 pullback of each arrow D1 → D2
is the cylinder of the corresponding hybridization map between the graphs.
Observe that the surface C associated to a diagram D (or rather the actual cylinder of the hybridization map) embeds into D̃ as a pullback of an
upstream path from D ∈ D to the source vertex. In particular, the pull52
backs of different upstream paths from D to the source vertex are mutually
homeomorphic.
The fibration D̃ → D linearizes to the (huge) commutative diagram of the
homomorphisms of the homology groups H1 (D = f −1 (D)), where D ∈ D,
with the obvious arrows corresponding to the edges D1 → D2 . Moreover,
each group H1 (D) comes along with the bilinear pairing (intersection of 1cycles in the surfaces) in the surface CD ⊃ D. The dimensions of these
groups as well as the (co-)ranks of the arrows and pairings are integer value
functions on the space D shaping its combinatorial personality.
Warning: if the collection S has a non-trivial symmetry (for example it
contains two strands of equal length with identical labels, or it contains a
circular strand where the labels have a non-trivial period, see Fig. 23) and
this symmetry persists under hybridization, one should distinguish between
a “diagram” and an “isomorphism class of diagrams”. In terms of D, diagrams with symmetries represent singular points where one should specify
the corresponding orbispace structure.
Global symmetries do not appear in biologically realistic situations but
partial symmetries do appear. In order to incorporate these in the categorical
formalism, we need a notion of a subdiagram and this can be achieved with
the universal moduli space of diagrams, D∗ = D∗ ( ∞ ) = D∗ ({A, G}), which
represents diagrams D(S) with variable sets S of strands, and injections
D1 (S1 ) D2 (S2 ) induced by embeddings S1 ,→ S2 .
Which way to go? Here (and this will be happening over and over again),
one faces a dilemma. One can pursue the intrinsic logic of the mathematical
construct and follow several mathematical avenues suggested by the space
D: one may investigate its relation(s) to the moduli spaces of Riemann surfaces, one can make a small category out of it and study the topology of
its classifying space, one can “complete” D by allowing diagrams on infinite
strands, one may generalize the notion of diagram by replacing the figure
∞ by a more general (Riemaniann) space (e.g. of negative curvature) or
by a suitable homomorphism of the fundamental group of the diagram into
another group, etc. Alternatively, one may concentrate on those aspects of
hybridization encoded in D, which appear more natural from a bio-chemical
standpoint, by trying to find biologically significant patterns in D, by developing computational means to analyze the structure of individual diagrams
as well as by introducing new structures on D’s and D suggested by bio53
physical considerations. The latter leads to new mathematical structures
and creates further forks along the road.
Temperature, entropy and Gibbs measures. We have already assigned energy to each vertex D ∈ D, and now we discuss how to assign a probability
(entropy) weight to a diagram D. This comes more or less naturally if we
“decorate” each D with the space realization of D, that is an embedding
Φ : D → R3 . Such a (D, Φ) is called a euclidean diagram. For our purposes, we consider only piecewise linear maps Φ sending edges of D to unit
euclidean segments. The “euclidian decoration” of the moduli space D has
three essential ingredients:
topology: the map Φ maybe “knotted”, namely there are many isotopy classes
of embeddings D → R3 and this can be recorded by augmenting the combinatorial structure of D.
energy: the combinatorial energy defined earlier can be made more precise
by taking into account the actual data on the intermolecular interactions
including a realistic bending energy as well as the hard-core repelling potential 43 that would automatically force Φ to be an embedding. Observe that
the bending energy is local on D, while the hard-core potential is local on
D × D rather than on D, which makes its analysis notoriously complicated
(as it reflects the excluded volume or self-avoiding property of Φ).
entropy: this refers to the total measure of the space E = E(D) of Euclidean
diagrams (D, Φ) as a function of D. The space E naturally embeds into
R3N , where N is the number of edges plus the number of vertices of D.
This space E is invariant under the isometry group of R3N , which makes it
infinite and so its measure should be normalized in a suitable way. If we
deal with self-hybridization diagrams (obtained by hybridization of a single
strand) we can pass to the factor E/R3 . But if disconnectedness of strands
and/or diagrams matters, then one should limit the range of Φ to a bounded
domain (corresponding to the tube where the solution of DNA is contained).
The next problem is that E ⊂ R3N is a (semi-algebraic) subvariety of positive
codimension and one should take special care on how to define the measure
on such a subset. One can use the natural induced piecewise Riemaniann
measure but it is customary to use the ambient Euclidean measure weighted
with the Gibbs factor coming from the energy. In any case, this measure
43
For two points x, y with d(x, y) ≤ , the “hard-core energy” equals, by definition, +∞.
54
is very difficult to evaluate and one resorts to some approximation to this
measure expressed entirely in terms of the combinatorics of D.
Given an energy function U : D(S) → R and measure weights µ(D) ∈
R+ , where D ∈ D = D(S), one is concerned with the pushforward ν = U∗ (µ)
of the measure µ under U . The measure ν on R is customary represented
via its Laplace transform G(β), that is the integral of the function e−βU over
R U with respect to ν. This equals the canonical sum
e−βU (D) µ(D)
G(β) =
D∈D
where each term µG (D) = e−βD µ(D) = e−U/T µ(D) is called the canonical
measure and where T = 1/β is interpreted as the inverse absolute temperature of the ensemble S of strands in the tube. The canonical sum carries the
same information as the specific heat capacity of the ensemble of our strands
in solution, which
as the
is an experimentally measurable quantity defined
2 T -derivative of d∈D U (D)µG (D). Clearly this derivative equals (T G (T )) .
This canonical sum, or rather its simplified versions, were extensively
studied for S consisting of pairs of complementary strands. In particular, by
looking at an Ising type approximation to G, one can make rather realistic
predictions of the melting behavior of DNA strands such as the zipping effect.
Also, by looking at G restricted to various segments of DNA one finds an
amazing matching with biologically significant patterns such as introns and
exons in genes, as well as coding regions for functional domains in proteins.
Another biologically significant case, concerns folding of individual strands
S of RNA, where “folding” refers to the measure/energy distribution on the
diagrams D(S). One is interested by how much this distribution for S’s found
in cells differs from that for random S’s.
The road to equilibrium. The Gibbs measure does not tell one how the
actual hybridization process develops but only describes the final equilibrium
stage of hybridization. Unlike the statistical ensembles usually studied in
physics, the relaxation time (i.e. the time needed to reach equilibrium) in
biology is relatively long, and the road to equilibrium maybe convoluted.
The non-equilibrium dynamics of ensembles of DNA’s can be modeled as a
random walk on D, by assigning suitable transition probabilities to the edges
between D’s in D. The assignment p12 to the edge D1 → D2 depends on the
difference between the Gibbs energies of D1 and D2 , and reflects to a large
55
extent the combinatorics of the folding (of the map D1 → D2 represented by
the edge), while p21 (which is not supposed to be 0 but usually smaller than
p12 ) reflects the combinatorics of the unfolding.
The major issue is the evaluation of the rate of diffusion of such a walk,
or/and of the first eigenvalue of the corresponding diffusion operator. To
get the flavor, we localize to the set of diagrams DL ⊂ D = D(S), where
the total number of double stranded edges equals L. Consider the graph
with vertices D ∈ DL , where the (non-oriented) edges between D1 , D2 ∈ DL
correspond to the diagrams of oriented edges in D of the form D1 ← D →
D2 , where D ∈ DL−1 . It seems not hard to evaluate the first eigenvalue of
this graph in terms of S and L, where the total length of the strands in S
and L go to ∞. For example, if S consists of a single strand S of length 2L
and NA = NA−1 = L then DL can be identified with the Cayley graph of
the permutation group SymL generated by transpositions. Actually, SymL
simply and transitively acts on the set of surfaces CD , where D ∈ DL , and
appears as a discrete approximation to the Riemann moduli space of surfaces
of variable genera. (This example does not incorporate the energy assignment
depending on the labeling and cannot tell us much about the random walk
we are truly interested in.)
Cleavage, ligation and emergence of stochastic symmetry. Let us bring
more chemistry into D∗ by introducing extra edges corresponding to creation
and cleavage of covalent bonds between strands. We write D1 ⇒ D2 if D2 is
obtained from D1 by ligating two ends of two (distinct or identical) strands
in D1 . Accordingly, D1 ⇐ D2 signifies breaking a strand at a single location.
We assign a weight to each arrow encoding the probability of the chemical
event represented by this arrow. To be realistic, we prescribe a relatively high
probability to ligation between two strands hybridized to a complementary
strand and having adjacent ends, as in Fig. 16. On the other hand, the
probability of ligation of non-adjacent ends is exceedingly low. Somewhat
unrealistically, we allow newly broken bonds to ligate again later on. (This
presupposes an influx of free energy activating the corresponding nucleotides.
Compare Fig. 1.)
We are interested in (random) populations of strands in a tube, where
the temperature oscillates between “high”, where the strands are completely
separated, and “low”, where strands conglomerate into diagrams. In the
latter case, strands with adjacent ends ligate with relatively high probability.
56
A
G
G
A
G
A
A
G
A
G
A
AG
G A AAGGGGAAGGGGAAAAA G G
G AAGGGGAAGGGGAAAA G
AA
GA
G
A
ligation site
G
A
G
A
A
G
A
AG
G A AAGGGGAAGGGGAAAAA G G
G AAGGGGAAGGGGAAAA G
AA
GA
G
A
Figure 25: Two strands hybridize with a third one and ligation takes place.
Cleavage happens at all temperatures but with smaller probability.
Question. Consider a random population of strands of various length,
i.e. that is a measure on the sequence space. How does this measure evolve
in time in the setting described above? Will strands with relatively high
repetition of patterns emerge?
Toy DNA symmetrization. To simplify, we work with the two letters A, G
and replace the Crick-Watson complementarity with the pairing AA and GG;
we emphasize ligation of two segments S1 , S2 which are fully hybridized to a
third one S3 such that their ends meet. We think about the subsegment S3
of S3 which is hybridized to S1 , S2 as a “ligation site” for S1 , S2 . See Fig. 25.
A population where subsegments of strands appear with high multiplicity have an advantage as, after cleavage, they have high chance to find hybridization sites. From this angle, the most advantageous (stable) population consists of pure A and pure G strands with a small percentage of
interbreeds, since spontaneous A-G ligation appears with probability which
is much smaller than spontaneous cleavage, while the A-A and G-G ligations
are competitive with cleavage due to the presence of many pure breed ligation
57
sites.
Amazingly, being random is sometimes advantageous for reconstruction
of cleaved strands, due to the high degree of specification: take a long strand
S and a strand S0 which is twin to a subsegment S0 of S. In the random
case, the binding energy between S0 and S0 is roughly twice as great as the
energies of other possible bindings of S0 to S, since the number of matches of
letters for random pairs of sequences is roughly equal to the number of mismatches. But for pure breed strands, the binding energy is constant for full
hybridization and, in general, proportional to the length of the hybridization
overlap. The same consideration applies to the hybridization of two pieces
of a cleaved strand on available ligation sites. Since the energy enters the
canonical sum under the exponent of the Gibbs-Boltzmann factor, it can
beat entropy and, depending on parameters (as temperature, energy, concentration, etc.), the population may evolve towards strands with repeated
random motives. One wonders if the tandem repetitions in genomes can be
explained by mechanisms of this nature.
What else is there? Mismatches of hybridization make some potential
hybridization sites unavailable. In particular self-hybridization of strands
(which is predominant at low concentrations) might “tie up” the ends of the
strands.
Since the sequence space, where the evolution of strands takes place, is
huge compared to the number of strands in the solution, one never arrives at
the true equilibrium state in real time. This means that the (limit) equilibrium distribution does not reflect the structure of an actual population at a
given moment of time. Instead of staying at equilibrium, one should rather
think of a relatively small cloud of strands (quasi-species in terms of Manfred
Eigen) wondering in the immense vastness of the full sequence space.
One may try to bring the above model closer to “real life” by taking complementarity and double strands seriously and by introducing other covalent
modifications of diagrams, such as recombination for instance. Besides, one
can prefer a large pool of random short activated nucleotides, rather than
allowing spontaneous reactivation of cleaved strands.
Actual polynucleotides in solution make a complicated ternary structure
not easily predictable by the self-hybridization pattern. Some of them display pronounced enzymatic properties, e.g. enhancing cleavage and/or ligation at specific sites of other strands. A possible way to incorporate this
phenomenon into the language of diagrams is, following Kauffman, to regard
58
the enzymatic activity as a random function on D∗ . Granted such a function,
it will significantly influence the evolution process favoring segments which,
after folding, promote ligation of self-like, and cleavage of competitors. For
example, pure breeds may be at a disadvantage similarly to pure strategies in
game theory. It would be interesting to realize this idea in a specific model.
Controlled paths in D∗ . There are several experimental techniques to
manipulate, control and analyze DNA in solution. These can be described
as operations on the space P of populations of diagrams, that are measures
on D∗ . Whenever we need random populations, we shall refer to the space
R. The operations are:
macroscopic compartmentalisation: instead of using one tube, one might have
solutions of DNA in n tubes, for relatively small n. This means that we deal
with n-dimensional vectors P = (p1 , . . . , pn ) ∈ P of diagram (in particular,
strand) populations, where each pi is a (random) measure on D∗ .
mixing solutions: the redistribution of the content of each of the n tubes in a
given proportion into m tubes, amounts to a specified stochastic n×m matrix
defining a linear map P n → P m . If the non-zero weights assigned by pi to
each diagram are sufficiently large (i.e. each diagram appears in sufficiently
many copies in solution), this formalism faithfully represents what happens in
the lab. But if some diagram appears in a small quantity, the number of them
going to a new tube, after mixing can be understood only probabilistically.
For example, if we have only one strand of a given species in a tube, we cannot
a priori tell in which of the tubes it goes after the mixing process. However,
a single molecule matters and can be detected a posteriori, for instance, with
amplification by PCR. Thus, we cannot round off small quantities and have
to keep track of them using the framework of random populations.
temperature control, hybridization and denaturation: raising temperature
to sufficiently high degree denaturates DNA and gives us an obvious map
from P = P(D∗ ) to the space of population of strands denoted by P(S).
Conversely, lowering the temperature, leads to hybridization, that is a map
P(S) → P(D∗ ). If the free energy of hybridization has a unique deep minimum (with a large attraction basin), then the hybridization is essentially
unique and non-ambiguous. In general, it is not so and the true map lands
in the space of random populations of diagrams. Also, the time profile of the
temperature may have significant effect on the result of hybridization. For
59
hybridization
Figure 26: Designed assembly of DNA strands. The bold traits (on the left)
correspond to the subwords designed to hybridize (on the right).
example, if one mixes certain partly hybridized diagrams, they may hybridize
along sticky ends arriving at a metastable state corresponding to a deep local
minimum of the free energy. After heating and slow cooling, one arrives at
the global minimum providing a population of maximally hybridized strands
(which may appear in different knotting/linking conformations).
Hybridization can be designed in order to serve (at least) two purposes:
the controlled self-assembly of a diagram out of strands and the realization
of non-deterministic algorithms. In the first case, one needs the free energy
with a unique sharp pronounced minimum, and in the second case one seeks
free energy functions with several deep (local) minima 44 with equal values
of the free energy. These two purposes are achieved by taking a collection of
randomly labeled strands modulo the following condition: there is a distinguished relatively small set of mutually complementary words which appear
as subwords in our strands. (See Fig. 26.) Then, the minima of free energy
are realized by diagrams hybridizing across these words. In both cases, one
starts with many copies of each strand. For self-assembly purposes, one needs
to purify the resulting population of diagrams in order to exclude undesirable
hybridizations. To realize a non-deterministic tree with N leaves, one needs
as many copies of each strand as is necessary to realize N different minima
with non-zero probability.
44
A specific RNA sequence admitting two distinguished foldings/two deep minima far
apart in the configuration space has been designed by Hervé Isambert with an experiment
being underway.
60
making primers: this means construction of specified vectors of measures in
P or in R, supported on the space of relatively short strands.
separation: this is a map s = sπ : P → P N , where π is a partition of P into N
subsets corresponding to the physical characteristics involved. The map sπ
consists of restricting a measure to each, among N , subsets of the partition.
The partitions π that we have in mind are those which can be biochemically
realized. For example, one can separate strands with gel-electrophoresis according to the length, or diagrams according to their mass and overall topology.
purification: it is the multiplication of a measure by the characteristic function of some subset in D∗ , interpreted as filtering away the diagrams where
the function vanishes. This should be physically realizable.
extraction: this is a map ex = exπ,r : P → P N , where π is a partition of
P into N subsets, r = {ri }N
i=1 are positive weights ≤ 1, and where the i-th
component of ex(p) equals p restricted to the i-subset of the partition times
ri . Formally speaking, extraction can be reduced to separation, purification
and mixing, but, in practice, a particular extraction may be feasible while
the corresponding separation and/or purification are not necessarily so. For
example, one can extract those diagrams which have a non-hybridized segment(s) with a given labeling(s) using complementary probes attached to a
solid support.
detection of p ∈ P: this refers to the evaluation of the integral p(d) of a
given function d : D∗ → R. We use those d’s where this evaluation is
experimentally feasible, preferably even for small values of p(d). An example
of this is provided by molecular beacons, where the function d take values 0, 1
depending on the presence or absence of a given subword in a strand, and the
intensity of luminescence equals the integral (average) of this (characteristic)
function.
changing connectivity of S: an enzyme is a map ei : D∗ → D∗ which breaks
or ligate strands at given locations, which are specific for a given enzyme. We
admit enzymes making several cuts within the same location (for example
EcoRI makes two cuts, as illustrated in Fig. 19). Thus every enzyme is
characterized by the size of the site at which it can work, by the combinatorics
and labeling of the site, and by the combinatorics of the site after the action
61
of the enzyme. Given ei , we obtain a map Ei : P → P by linearly extending
ei .
disactivation of ligation: one can disactivate the ends of the strands in a given
tube (compare to the paragraph on Synthesis of polynucleotides in Section 5)
and thus prevent the strands from ligation. The disactivation is a labeling
of certain ends of strands that allows a design of a wider variety of ligation
maps ei .
complementation: given a diagram, one can generate from it the complements
to all non-hybridized segments using some replicase enzyme in the presence
of an excess of free nucleotides. This amounts to adding the formal sum of
these subsegments to the δ-measure supported on this diagram.
Also, one may generate a specific subsegment of each non-hybridized segment, starting from a given short subword in it and using a primer complementary to this subword together with DNA polymerase enzymes. This
amounts to an operation on populations similar to the above.
amplification: repeatedly applying complementation and denaturation leads
to measures on D∗ having an exponentially large weight on specific strands.
In particular, with replicase, one can multiply a given measure, symmetric
under complementation, by 2n in n steps.
Granted these operations with specified parameters for the free energy
and error margins expressed in the degree of randomness of the arising diagrams, one poses two questions:
- can one faithfully assemble a particular diagram? Here, one is additionally interested in a diagram with a prescribed embedding into R3 , e.g. in
the realization of specified DNA knots. (Many classes of single and double
stranded knots have been already chemically implemented.) For this, one
has to modify the above formalism by incorporating in it the combinatorics
of the double helical structure of DNA;
- which (non-deterministic) computations can be modeled by the above operations on population of diagrams? For example, given two populations
p1 , p2 , how many operations one should perform over p1 in order to obtain
p1 approximating p2 with a prescribed error?
Lesson from D∗ . Our formal description of DNA is by no means final neither from a bio-chemical nor from a mathematical standpoint. It is
62
not sufficiently specific for practical applications; yet, it illustrates some of
the general principles mentioned earlier: complementarity and templating,
self-assembling where hybridization and denaturation serve as scaffolding for
making/breaking covalent bonds, repetitiveness of stochastic motives, channeled relaxation via designed free energies.
On the mathematical side, one seeks for a more general class of models
incorporating random walks on combinatorial moduli spaces and the design
of (free energy) functions with specified symmetry of local minima with an
explicit separation of several (at least three) scales. (In the D∗ -model the
weak bonding serves as intermediate between thermal energy and covalent
energy, where the concentration of activated bases appears as “covalent temperature”.)
Self-assembly.
Tilings. Consider a connected subset T (tile) in R3 , for example a convex
polyhedron, with a distinguished subset of mutually complementary (possibly overlapping) non-empty domains on the boundary, denoted Db , D̄b ⊂ ∂T ,
where b runs over a (possibly infinite) set B. We are interested in assemblies generated by T , that are subsets A in the Euclidean space, decomposed
into a union of congruent copies of T where two copies may intersect only
at their boundaries and have a “tendency” to meet across complementary
domains on the boundary. We have in mind a protein molecule T with complementary active sites Db , D̄b such that different copies of T bind along the
complementary domains and self-assemble into complexes. In the geometric
context we specify the binding properties by introducing (binding) isometries b : R3 → R3 to each b ∈ B such that T and b(T ) intersect only at the
boundary, and b(Db ) = D̄b . From now on B is understood as a subset in the
Euclidean isometry group Iso(R3 ).
Accordingly, we define an assembly A associated to (T, B) by the following
data:
- a connected graph G = GA with the vertex set 1 . . . N ,
- subsets Ti in R3 , where i = 1 . . . N , which may mutually intersect only at
their boundaries,
- an isometry bk,l : R3 → R3 moving Tk onto Tl , for each edge (k, l) in G,
such that there exists an isometry ak,l which moves Tk to T and conjugates
63
bk,l to some isometry b in B. Notice that this b is uniquely determined by
bk,l up to conjugation.
Several tiles. If we start with several different tiles T 1 , . . . , T n rather
than with a single T , we consider the sets of pairs of binding isometries
i,j
i
j
B i,j ⊂ Iso(R3 )×Iso(R3 ) such that bi,j
1 (T ) and b2 (T ) intersect only at their
boundaries and their intersection is non-empty. The definition of an assembly
associated to ({T i }, {B i,j }) goes as above with the following modifications:
the graph G has vertices colored by the index set 1 . . . n, the corresponding
subsets in R3 are denoted Tki where i = 1 . . . n and k = 1 . . . Ni , and finally, we
forfeit the isometries bk,l and for each edge (k i , lj ) we emphasize an isometry
j
i,j
i
j
of R3 which moves Tki to bi,j
1 (T ) and Tl to b2 (T ).
In what follows, we sometimes refer to the union of tiles defined above,
as an assembly.
Qualities of an assembly. The tightness of the tiling is one quality that
chemists appreciate. This can be measured by the number of cycles in the
graph G, or equivalently by the minus Euler characteristic of the graph.
The imperfection of a tiling is measured by the “unused” areas of the
boundaries of the tiles. First define the active domain ∂act (T ) ⊂ ∂T as the
union of the intersections of ∂T with b(T ) for all b ∈ B. Then define the
“unused boundary” ∂un (A = ∪Ti ) as the union ∪N
i=1 ∂act (Ti ) minus the union
of the pairwise intersections ∪(k,l)∈G Tk ∩ Tl . An assembly is called perfect if
the area of the imperfection equals zero. We say that an assembly contained
in a given subset X ⊂ R3 is perfect with respect to ∂X, if ∂un (A) ⊂ ∂X.
The uniqueness refers to the uniqueness of an assembly subject to some
additional constraints. For example, given an X ⊂ R3 , one asks first if X can
be tiled by (T, B) and then asks for the uniqueness of such a tiling. We say
that (T, B) generates an unconditionally unique assembly if every imperfect
assembly uniquely extends to a perfect assembly.
The essential problem of tiling engineering is designing a relatively simple tile or a few such tiles which assemble with high quality into large and
complicated subsets in R3 . Here is a specific example for the unit sphere
S 2 rather than S 3 , where one uses the obvious extension of the notion of
tilings to homogeneous spaces. Given 8, δ > 0, consider triangulations of the
sphere into triangles ∆ with Diam(∆) ≤ 8 and area(∆) ≥ δDiam2 (∆). It
is easy to see that the number of mutually non-congruent triangles in such
64
Figure 27: Vernier. Rodlike tiles differing in length form an assembly that
grows until the ends exactly match.
template
assembly
binding sites
Figure 28: A tile is stable in the assembly only if it binds at two adjacent
binding sites. The stability of the whole assembly is insured by the enforced
stability of the template. The formal description of this example is not
completely captured by our model.
a triangulation, call it n(8, δ), goes to ∞ for 8 → 0 and every fixed δ > 0.
The problem is to evaluate the asymptotic behavior of n(8, δ) for 8 → 0 and
either a fixed δ or δ → 0.
Real life examples. It remains unclear, in general, how cells control the
size of (imperfect, with some unused boundary,) assemblies, but certain mechanisms are understood. For example, out of two rod-like molecules of length
three and five, one gets a double rod of length 15 as illustrated in Fig. 27.
Another strategy is starting an assembly from a given template (see Fig. 28
Figure 29: Polymeric structure growing until the energy required to fit new
subunits becomes too large.
65
new binding site
Figure 30: Tiles which differ in shape and binding sites. Their binding
generates a new contiguous binding site.
for a specific design). Sometimes, tiling is non-isometric: tiles slightly bend
in order to fit, and the assembly terminates when the bending energy becomes too costly or when the accumulated bending deforms and disactivates
the binding sites (see Fig. 29). Also, the binding of a ligand to an active
site might change the shape of the molecule and thus influence the binding
activity of other sites. Another possibility is the creation of a new binding
site distributed over two or more tiles bound together on an earlier stage of
the assembly (see Fig. 30).
These mechanisms may produce a non-trivial dynamics in the space of
assemblies in the presence of free-energy. (The delivery of free-energy into
a self-assembling system is hard to realize bio-chemically. Even mathematically, modeling such processes does not seem easy.) In particular, one may
try to design a periodic motion of a tile over a template, something in the
spirit of a RNA-polymerase cycling around a plasmid.
Programmed design. Can one find a small set of relatively simple tiles such
that, starting from a template supporting a linear code (that may be a DNA
or RNA molecule incorporated into a macromolecular complex), the assembly
process will create a given three dimensional shape in the space? We think
here of interacting tiles performing a transformation from labeled templates
into three dimensional structures and we ask what kind of transformations
can be realized in this way. Also, we want to understand how much the
complexity of the construction depends on the complexity of the tiles, where
the latter can be measured by the number of the binding sites of the tiles,
the size of the sets B i,j , etc.
66
point s of S
δ
δ
S
disc with radius ε
Figure 31: The schema of a rod lying along a surface S (left), and the section
of a liposome, a bi-layer surface packed with rod-shaped molecules (right).
Liposomes and minimal surfaces. Consider a smooth closed surface S ⊂
R and let Sδ denote a small normal δ-neighborhood of S, that is the union
of the 2δ-segments [−δ, δ]s ⊂ R3 , over all s ∈ S. We want to think of Sδ
being tiled by the segments [−δ, δ]s and to make this realistic we replace the
segments by thin rods as follows: pack the surface S by N 8-discs Dsi ,% ⊂ S,
where i = 1 . . . N , si ∈ S and where 8 is much smaller than δ; let lsi ,%,δ be
the union of the segments [−δ, δ]s over s ∈ Dsi ,% . The union L = LS,%,δ =
∪N
i=1 lsi ,%,δ is called the micelle assembled out of the rods lsi ,%,δ .
Such “micelles” appear in nature, in particular in cellular membranes,
where the major structural component is a phospholipid bi-layer built of rodshaped molecules having one hydrophobic and one hydrophilic end 45 . Such
3
45
Water molecules are polarized and interact with each other via rather strong hydrogen
bonds. If an organic molecule has a comparable hydrogen interaction with water molecules,
a distribution of such molecules in the water does not increase, or even decrease, the
overall energy of the system. Such molecules are called hydrophilic and the corresponding
substances, such as sugar for example, easily dissolve in water. Other molecules M (such
as lipids, or benzene for instance), called hydrophobic, do not interact with water due to
their spatial geometry and especially to a particular (non polar) distribution of the electric
charges. In this case, insertion of M ’s into water increases the energy of the system as it
prevents the water molecules adjacent to M ’s to interact with each other. This creates an
67
bi-layer surfaces, called liposomes, can assemble spontaneously in a solution
containing phospholipids. See Fig. 31. The shape of a liposome is determined
by the equations governing hydrophobic forces which transform, in the limit
for 8, δ → 0, to specific partial differential equations on the resulting surface
S. For example, the membrane of an erythrocyte (i.e. red blood cell) is
believed to minimize its integrated squared curvature with a given area, and
enclosing a given volume 46 . This suggests the following:
Question. Given an isotopy class of closed surfaces in R3 does there exist
a surface S in this class with a given area and a given enclosed volume which
minimizes the integral of a given function of principal curvatures? (The most
studied case concerns the integrated squared mean curvature as it enters
the Willmore conjecture; other integrants were not apparently investigated
much.)
Tilings, energies and variational equations. Let us encompass the above
into a more general scheme incorporating energy into the idea of assemblies
of tiles.
Let V be a smooth manifold (V = R3 in the previous discussion) and let
P → V be a smooth fibration associated to the full frame bundle F r → V .
We think of the fiber Pv ⊂ P, for v ∈ V , as the configuration space of a
tile (a protein molecule) in V located at v. In fact, the position of a tile
in the Euclidean space is uniquely determined by the location v ∈ R3 of its
center of gravity and an orthonormal frame at this point, that is a 3-tuple
of orthonormal tangent vectors at v. Similarly, a phospholipid molecule is
modeled by a tangent vector in R3 .
The configuration space for N identical tiles is the Cartesian power P N ,
where the rules of tiling are encoded into a (energy) function U : P N → R. A
apparent attractive force between hydrophobic molecules in water, which is proportional
in energy to the area of the boundary between water and the M -substance. Thus, minimal
energy is achieved by a minimal area per given volume: a small amount of M -substance in
water, or water in M takes spherical shape. Large molecules, such as proteins, might have
both hydrophobic and hydrophilic parts (some amino-acids are hydrophobic and some
are hydrophilic) and so when they fold, they expose most of their hydrophilic part on the
surface in contact with water. Liposomes try to minimize (and reduce to zero as illustrated
in Fig. 31) the area of their hydrophobic surface.
46
This is assumed by the bio-medical community but apparently there is no mathematically rigorous proof. The main technical point is to establish the axial symmetry of the
solution of the variational problem.
68
distinguished class of energies is given by the two-particle interactions, that
are functions u : P 2 → R, with U defined in the usual way:
U (p1 , . . . pN ) =
N
u(pi , pj ).
i,j=1
Example. In the language of the paragraph on Tilings (see earlier in
this section), the function u is defined on the pairs of frames (p1 , p2 ) in R3
representing pairs of tiles (T1 , T2 ) as follows:
- if the interiors of the tiles T1 and T2 intersect, then u(p1 , p2 ) = +∞.
- if T1 intersects T2 across binding domains on their boundaries defined with
b ∈ B, then u(p1 , p2 ) = u0 (b) for a given negative function u0 : B → R
recording the corresponding binding energy.
- if the tiles T1 and T2 are disjoint then u(p1 , p2 ) = 0.
Observe that this potential is singular and discontinuous. A natural class
of potentials is constituted by those which are smooth on strata of some
stratification of the configuration space. Often, one needs to regularize them
and keep track of the fine local geometry of the stratification.
One can define tilings/assemblies as local minima of U . Everything we
said about tilings before can be reformulated in this language. Physically
speaking, this (local) minima correspond to (meta)stable states at zero temperature. In order to bring in positive temperature, we need a canonical
measure on the space P and thus, on the space P N . Usually, for instance
for frames and rods in the Euclidean space (and in Riemaniann manifolds V
in general), there is a non ambiguous canonical measure invariant under the
Euclidean isometry groups. In most cases, the choice of the canonical measure is influenced by the Louiville measure on the phase space “overlying”
PN .
Given a canonical measure dµ, one can speak of Gibbs states (measures)
−βU
dµ and model an approach to equilibrium (Gibbs) states by biased rane
dom walks in P N . The quality of a tiling can be now expressed as the rate
of approach to equilibrium and the size(s) of the attraction basins of various states. Much of the discussion on population of diagrams extends to
population of tilings.
Minimal surfaces. Given a class of subvarieties W ⊂ V distinguished by
a variational equation, one wants to obtain them as limits of Gibbs states
69
with suitable U on P N , for N → ∞, with simultaneous rescaling of U . Is it
always possible? Can one go away with two-particle potentials? (The idea
is to divide a minimal surface into “particles” corresponding to infinitesimal
elements, dissolve them in a tube and then reassemble them, following a biased random walk guided by some energy function between the particles. We
want the reassembled object to satisfy the original partial differential equations.) The latter seems unlikely in view of the fact that the hydrophobic
forces shaping liposomes are not two-particle 47 . (If one thinks of the limits
of N -tuples {pi } as measures on P, one is lead to a generalization of Almgren’s theory of varifolds where the potential is not reducible, or at least not
immediately, to interactions between finitely many particles.)
Bending, curvature and multi-jets spaces. In order to incorporate bending
and the resulting energy of molecules, one can generalize the above discussion
by replacing F r by the k-jet frame bundle F rk which, for example, encodes
curvatures for k = 2. This suggests looking at higher order variational equations, where basic examples are provided by complex algebraic geometry.
Given a complex submanifold W of dimension m in a Kähler manifold
V , one can associate to it submanifolds Wk,N in larger Kähler manifolds by
taking N -tuples of suitably reduced k-jets of germs of W in V . Each of these
Wk,N is volume minimizing in the ambient (complex) space (which comes
with a natural Kähler metric), and thus the original W satisfies a hierarchy
of multi-differential equations. Can these equations be recaptured from appropriate Gibbs states? How much of the complex analytic nature of W is
reflected in the (diffused) Gibbs states? Probably one needs here a multiscale
limit where the energy comes on different levels (weak bonds, covalent bonds,
etc.) and takes care of different derivatives and/or singularities.
Complementarity, affinity and protein design.
An approximate complementarity relation on a set P (which may be a set
of proteins or of macromolecules in general) is a positive symmetric function
a : P × P → R, where a(p1 , p2 ) measures a certain affinity between p1 and
p2 . One thinks of the relation A% = {(p1 , p2 )|a(p1 , p2 ) ≥ 8} ⊂ P × P as a
47
Probably, one can formally reduce hydrophobic forces to two-particle interactions by
explicitly introducing the water molecules into the picture. Here, one needs to go to the
double limit, size(H2 O)/δ → 0 and δ → 0, expressing the idea that the water molecules
are small compared to the size δ of phospholipids.
70
Figure 32: Protein complexes formed by binding of complementary rigid surfaces (left), by helix-like coiling (center), and by binding of a long polypeptide
chain with a rigid surface (right).
multivalued map A% : P → P and one wants to express the idea of this map
being an approximate involution. This is done with the inequality
(∗)
a(p1 , p4 ) ≥ F (a(p1 , p2 ), a(p2 , p3 ), a(p3 , p4 ))
where pi ∈ P , for i = 1 . . . 4, and F = F (a1 , a2 , a3 ) is a positive symmetric
function monotone increasing in each variable (chosen accordingly to a given
situation). Such an F is supposed to capture the idea of the approximate
key-lock binding mechanism between proteins and other (macro)molecules.
See Fig. 32 for possible geometric patters of binding mechanisms.
Examples. Let P consist of the polynucleotides of a given length l and
let a be the energy of the best hybridization between them, where we assume that the energy of the hybridization equals the number of mutually
complementary pairs at the corresponding positions j = 1 . . . l. We write a
polynucleotide as a string p(j) and agree that p1 (j) · p2 (j) = 1 if the bases
p1 (j) and p2 (j) are complementary, and 0 otherwise. We express the hybridization energy as
a(p1 , p2 ) = p1 , p2 =
l
p1 (j) · p2 (j).
j=1
Clearly, (∗) is satisfied here with F = max{0, a1 + a2 + a3 − 2l}. This
is meaningful only for large a’s, namely ≥ 2/3. On the other hand, the
affinity between two random strands equals 1/4 of l. Furthermore, if pi , for
i = 1 . . . 4, are random strings conditioned by the inequalities a(pi , pi+1 ) ≥ ai ,
71
for i = 1, 2, 3, then there is a better (easily computable) lower bound on
a(p1 , p4 ). This suggests that it may be more useful for practical problems
(see below) to restrict a to random points p ∈ P where (∗) is satisfied with
a greater F .
Design of antibodies. Given p0 ∈ P , one seeks p ∈ P with possibly large,
ideally maximal, value of the affinity function a(p0 , p). In practice, p0 is a
given macromolecule, e.g. a protein, and one looks for a protein p which
fast and strongly binds to p0 . “Strongly” usually refers to the dissociation
time Tdis between p and p0 , that is the half life of the association between
p and p0 . “Fast” is indicative of the association time Tass , i.e. the average
time needed for forming the bound state from the unbound one. In practice,
one does not work with only two single molecules p0 and p but rather with
ensembles of them in solution, and the times Tdis , Tass are measured per given
concentrations.
Ideally, one wishes an antibody to be both strong and fast. Pharmacologically, one is happy with p with a relatively short (but not too short)
dissociation time and the shortest possible association time. (Early immune
response is crucial for fighting an invader.)
Let X be the space representing pairs of molecules p0 and p in solution
(where the space scale is adjusted to implement a realistic concentration of
ensembles of molecules in solution: two particles ask for a very small tube!).
The ratio of the times Tdis /Tass can be expressed in thermodynamical terms
as the ratio C∝ of the Gibbs measure on X of the bound states (po p) and
the Gibbs measure of the unbound states (p0 p)
Tdis /Tass = C∝ = µ(X)/µ(X),
since the space average equals the time average. This ratio depends on the
binding energy of p0 to p and can be, in principle, computed provided that
we know the geometry/energy profiles of the binding sites of the molecules
p0 and p.
The second relevant thermodynamical invariant, the average transition
rate R∝ , refers to the “area” of the common boundary X∝ between X
and X in X. This depends not only on the Gibbs measure but also on
the dynamics in the space X describing the time behavior of the molecules
and equals the average number of intersections of time orbits with X∝ . An
alternative geometric definition is 8−1 times the Gibbs volume of the 8-orbit
72
of X∝ , for 8 → 0. (This makes sense for stochastic as well as deterministic
time behavior and we shall return to this later.) The times Tdis and Tass are
expressible in terms of C∝ and R∝ as follows:
Tdis =
C∝
(C∝ + 1) · R∝
and
Tass =
1
(C∝ + 1) · R∝
The theoretic determination of R∝ for proteins is more difficult than that
of C∝ because it depends on the effects of weak affinity over the whole surfaces
of the proteins, and not only at their binding sites. On the other hand, the
times Tdis and Tass can be measured experimentally with essentially equal
ease (or difficulty).
The affinity function a extends by bilinearity to the space Π of populations
of proteins, that are measures π on P . The experimental techniques provide
a bonus of measuring a(π1 , π2 ) essentially with the same ease as a(p1 , p2 ):
given two populations π1 , π2 in two tubes, one mixes the contents of these two
tubes and isolates the pairs of proteins which stuck together. In practice, the
proteins from the population π1 , that are the antigene(s) for which we seek
an antibody, are attached to a solid support, and used to extract antibodies
p’s from π2 that are free in the solution. The latter are the antibodies which
are displayed for example on the coating of viruses (as mentioned at the end
of Section 5). This allows one to measure not only the value of a(π1 , π2 ) but
also to identify specific proteins in the populations π1 , π2 with mutual high
affinity.
The computational power of such an experiment is quite impressive: with
105 proteins on the solid support and 105 in the solution (these amounts are
realistic) one achieves a selection among 1010 possible pairs of proteins.
This procedure is repeated many times using artificial evolution of viruses
selected according to affinity of the corresponding antigene(s). Mathematically speaking, one maximizes the function a(p0 , p) by the following algorithm: start with an arbitrary p which hopefully gives a relatively large
value to a(p0 , p), and then take a population of small (≈ 105 ) variations pi of
p. Choose among the pi ’s, the one with the largest a(p0 , pi ) and repeat the
procedure with this pi instead of p.
One can make this process more efficient with the same computational
effort by varying p0 as well as p. For example, at the i-th step one can
replace p0 with a population π0 (i) of proteins, where π0 (i) converges to p0
for i → ∞, and apply the i-th step to a(π0 (i), p). This is a schematization
73
of the approach being currently implemented by Bill Huse in collaboration
with Michael Freedman (personal communication by Huse) 48 . One wonders
whether the approximate complementarity of the antibody/antigene interaction expressed by (∗) can be of any use for protein engineering.
States, dynamics and relaxation.
There are several dialects in the language of dynamical systems. We
shall limit ourselves to three of them: point dynamics, fuzzy dynamics and
stochastic dynamics. To get the idea, start with a Riemaniann manifold V
where:
- point states are tangent vectors and the dynamics is given by the geodesic
flow;
- fuzzy states are subsets U ⊂ V and the image after the dynamics t-map is
the t-neighborhood of U , that is the set of point in V within distance ≤ t from
U . Alternatively, one may restrict to the boundary ∂U transformed to the
boundary of the t-neighborhood of U , following, apart from the wavefront
singularities, the geodesic flow applied to Legendrian submanifolds in the
unit tangent bundle.
- stochastic states are (probability) measures on V and dynamics is the common diffusion semi-group defined by the heat kernel.
Following the classical physics paradigm, one looks for point dynamics
describing real life systems. If necessary, the space is enlarged by introducing extra “hidden parameters”. In the cell, however, the “complete states”
carrying the full information for the time development, are apparently not
points but probabilities on the configuration space X of atoms and molecules
constituting the cell. (It seems unreasonable to keep track of momenta of
particles; some quantum parameters of the molecular states though, may be
relevant for the functioning of the cell.)
The space X is too large and too fine to be directly observable and what
one sees in experiments are states in some reduced spaces which are certain
quotient spaces of X 49 defined by classes of phenomenological observables.
This cannot be explained without resorting to formal definitions.
48
Another context where the idea of introducing parameters appears is the Shub-Smale
algorithm for finding zeros of complex polynomials.
49
There are other mechanisms of reduction besides naı̈ve quotients, such as the symplectic reduction for example, but these lie out of the scope of the present article.
74
A point predynamics on X is a map A : X × T → X, where T is the
time domain which is usually either R or R+ . It might become a more
general semi-group, if needed, such as Rn for Cartesian products of n systems
with a possible symmetry reduction due to permutations between identical
particles 50 .
A fuzzy predynamics on X is a map A with X replaced by the space P(X)
of subsets of X, or by some “reasonable” subspace in P(X), e.g. measurable
subsets, closed subsets, semi-algebraic subsets, Lagrangian and Legendrian
submanifolds, etc. One usually requires A(X1 ∪ X2 , t) = A(X1 , t) ∪ A(X2 , t)
and A(X1 ∩ X2 , t) ⊂ A(X1 , t) ∩ A(X2 , t), for X1 , X2 ⊂ X.
Similarly, linear stochastic predynamics is defined with the space M(X) of
measures on X instead of P(X), where we usually restrict to Borel measures
µ on topological spaces X (or to smooth measures on manifolds X) and we
require A to be linear in µ ∈ M(X).
A point predynamics on X naturally induces the fuzzy and the stochastic
predynamics.
A predynamics A is called dynamics if it satisfies the semi-group property:
A(A(x, t1 ), t2 ) = A(x, t1 + t2 ).
Example. Let X be a metric space and A be a fuzzy predynamics, where
A(Y, t), for Y ⊂ X, equals the t-neighborhood of Y . If X is a Riemaniann
manifold or an arbitrary length space for this matter, then this predynamics is a dynamics. Conversely, if this A is a dynamics then X is a length
space (modulo trivial readjustment). Thus, a fuzzy dynamics structure on
a space generalizes the (non-symmetric) length metric structure. (One can
meaningfully generalize further, by allowing a multi-dimensional T . Compare
footnote 50.)
The Gauss’ shortest path construction of the intrinsic metric out of the
extrinsic one, suggests how to go from a predynamics A to a dynamics A in
general:
(i)
−1
A (x, t) =def lim A (x, i t).
i→∞
(i)
Here, A stands for the i-th iteration of A, and we apply this definition to
T = R+ whenever the limit exists. (In the physical tradition, every limit
exists unless otherwise proven.)
50
The classical geometric situation of “interesting” time presents itself in the Weyl chamber flows over real and p-adic locally symmetric spaces.
75
Involutive dynamics. A point dynamics is called involutive 51 (time reversible) if there exists an involution on the space X reversing the direction
of the dynamics
A(Invo(x), t) = Invo(A(x, −t)).
For example, the geodesic flow is obviously involutive as well as the system
of classical mechanics described by second order differential equations.
A fuzzy dynamics is involutive if
(A((A(Y, T ))⊥ , t))⊥ ⊃ Y
where Y ⊂ X and ⊥ is the operation of complementation of subsets of X.
Riemaniann geometry is involutive as the metric is symmetric. The nonstrictness of the displayed inclusion is due to the presence of wavefront singularities.
A stochastic dynamics is involutive if the infinitesimal generator ∆ of A,
provided it exists, is a symmetric operator. This is the case, for example, for
the diffusion on Riemaniann manifolds where the symmetry of the (Laplace
operator) ∆ results from the symmetry of the metric. Similarly, in the chemical kinetics, the involutive property (detailed balance) of the underlying point
dynamics implies that for the stochastic dynamics (Onsager relations).
Reduced dynamics. Given a surjective (factorization) map f : X → V , we
want to derive a dynamics on V , representing the space of phenomenological
states, from a given A on X. Every fuzzy (in particular, point) dynamics
on X obviously projects to the fuzzy predynamics on V . Sometimes it needs
to be transformed to a dynamics by the above procedure, but sometimes
it comes for free. (As it happens, for example, to the geodesic dynamics
projecting from the tangent bundle to the underlying Riemaniann manifold.)
The reduction of stochastic dynamics needs distinguished (canonical)
measures, one on X and one on V 52 , so that certain measures, called smooth,
are representable by density functions. Such a function can be lifted from
V to X, turned into a measure, transformed by A and then pushed forward
back to V . This brings down a stochastic dynamics from X to V under a
suitable smoothing effect of A on measures.
If we start with an involutive point dynamics on X and the map f is
involutive (i.e. f (Invo(x)) = f (x)), then the resulting fuzzy dynamics is
51
52
Flows of Weyl chambers suggest larger reflection groups compatible with dynamics.
The measure on V typically comes as the push-forward of that on X.
76
involutive. This is also true in the stochastic case if (in the agreement with
the Onsager relations) the point dynamics preserves the canonical measure
and this measure pushes forward to V .
In some cases, the full space of point states can be reconstructed from its
phenomenological reduction(s) V in a meaningful way. For example, if V is a
Riemaniann manifold or a general length metric space then one may take the
space X of all locally isometric maps from R to V with the obvious action
of T = R. Clearly, the symmetry of the metric make the geodesic dynamics
involutive. If the space V is smooth (or more generally, of curvature bounded
from below), then the geodesics cannot branch. This corresponds to causality
in a physical situation. On the other hand, the branching of geodesics at the
point of infinite negative curvature is reminiscent to the decay of a molecule in
the course of a chemical reaction. The latter process seem to need an infinite
dimensional space X even if we start with a finite dimensional space V , due to
infinite repetitions of the ambiguities involved in the decay process. Here is a
construction of X showing this phenomenon. Let A1 , A2 be point dynamics
on spaces V1 , V2 respectively, and let W be a multivalued correspondence
between V1 and V2 which is given by a pair of “fibrations” W → U1 ⊂ V1 and
W → U2 ⊂ V2 . Define X as the space of orbits constructed as follows: take
a point v ∈ V1 ∪ V2 , say in V1 , and follow it by the A1 dynamics until its first
entry u1 ∈ U1 ; then take a point u2 ∈ U2 corresponding to u1 , which means
that u1 and u2 come from the same point in W . Continue by applying A2 to
U2 , until the orbit returns to U2 and repeat the same process indefinitely.
This construction makes sense in a variety of contexts (measurable dynamics, piecewise smooth dynamics, etc.) and it models the chemical associations/dissociations of molecules when V1 represents the configuration (phase)
space of molecules entering the reaction and V2 describes the product of the
reaction, while W corresponds to the short life transition states.
In many biological situations, the phenomenological reduction of the “universal” space X is too drastic to recapture X. The purpose of molecular
biology is to find sufficiently many observable parameters needed for reconstructing the point dynamics on X.
Isoperimetry, Cheeger constant and the transition rate. Let X be a space
with a fuzzy dynamics A and a distinguished measure µ on X. Define
mA (µ0 , t) = µ(A(X0 , t))
77
for X0 ⊂ X and µ0 = µ(X0 ). The class of functions mA = mA (µ0 , t) is called
the isoperimetric profile of A, where the essential information is encoded by
the Cheeger “constant”
Ch(µ0 , t) =
inf
X0 ⊂X,µ(X0 )µ0
mA (µ0 , t) − µ0
t · µ0
The basic invariant of a stochastic dynamics is the rate of approaching
equilibrium (transition rate) which can be expressed in terms of the first
eigenvalue λ1 of the infinitesimal generator of A. In geometry, λ1 can be
bound from below in terms of the Cheeger constant and this idea extends to
more general stochastic dynamics, in particular those obtained by reduction
of point dynamical systems.
The rate of relaxation (transition) problem relevant to biological systems
can be formulated either in the point dynamics framework for a given class
of phenomenological observables, or in terms of the stochastic dynamics of a
reduced system. In the former case, we start with a linear subspace Φ in the
space of all functions (observables) on X and we look for an upper bound
on a suitable norm (e.g. the L2 -norm) on Φ in terms of a similar norm on
t−1 ·(φ−A(φ, t)), for φ ∈ Φ, and for the action of A on observables induced by
the action on X. If φ is a characteristic function of a subset Y ⊂ X, such a
bound can be thought of
as an isoperimetric inequality, with the 8-boundary
of Y defined as ∂% (Y ) = 0≤t≤% A(Y, t) ∩ Y .
In general, there is no non-trivial bound 53 of the measure µ(Y ) by 8−1 ·
µ∂ (Y ) , for (infinitesimally) small 8, but such bounds are expected for many
particular observables, especially those lifted from sufficiently “strong” and
“regular” factorizations V of X. Here, “strong” means that the fibers of the
map f are sufficiently large, e.g. they have a relatively low codimension and
they are predominantly transversal to the orbits, while “regular” may refer
to the regularity of the map f as well as to the invariants of f under some
symmetries of the space X.
There is a variety of techniques in geometry for proving isoperimetric
inequalities, where many of them are based on the variational techniques reducing the general problem to the “worst” case in a given class of observables
(or domains Y ⊂ X) and then showing that the worst case is not so bad after
53
Such bounds are possible for non-amenable measurable actions: thy are always present
for actions of Kazhdan’s T-groups.
78
all. In science, in order to bound from below the rate of transition one looks
for an explicit realization of the intermediate states allowing fast transitions.
Example: hybridization. We mentioned earlier in this section how hybridization proceeds in steps via the zipping mechanism ensuring a high
transition rate between fully dissociated and fully hybridized states.
Example: regulatory proteins. The association time for a regulatory protein finding the regulatory region on DNA is reduced by a certain affinity
of this protein with DNA all along the strand. The association process is
divided in two stages: first, the protein finds DNA by randomly moving in
the solution. The resulting weak binding of the protein to the DNA takes
significantly less time than finding the regulatory region on random, since
the size of DNA is by far larger than that of the regulatory region. Next, the
protein starts a random dance in the vicinity of DNA and eventually finds
the regulatory region. This random dance on DNA is more time efficient
than a random walk in the solution since the size of DNA is small compared
to the volume of the cell.
We have also seen intermediate energy scaffolding in the self-assembly
processes as well as in the binding of antibodies to antigens. The scaffolding
can be justified by a simple computation showing the drastic reduction of
the association time within the range of parameters actually observed in the
cell, where the relevant parameters in the case of regulatory proteins are:
- the energies of the weak and strong (at the regulatory region) bindings,
the thermal energy and the related rate of diffusion (the random walk in
solution);
- the volume of the cell, the size of DNA and of the size of the regulatory
region, all of them measured in entropic terms.
The above can be interpreted in terms of parametric quantitative ergodic
theory, where one is concerned with the rate of convergence R of the time
average of φ ∈ Φ over the interval [0, t], where the space Φ as well as the
dynamics A depend on auxiliary parameters, say τ and θ. For example, τ may
signify the magnifying power of an observational device, and the dependence
of A on θ may encode the ratio between the energies which goes to infinity
for θ → ∞. The problem consists in evaluating R for the parameters
t, τ, θ → ∞ in a certain coherent way, where the dynamical system and the
space of observables may degenerate in the limit.
79
Such an ergodic theory, incorporating the variational isoperimetric techniques, may prove helpful for finding transition states (scaffolding) enhancing
relaxation in cells (e.g. protein folding), and for suggesting numerical algorithms modeling the relaxation process. A more ambitious goal concerns the
fundamental transitions from non-life to life, where one may start with developing evolutionary realistic scenarios of replication of dynamical systems.
Population states and population dynamics. In many physical, chemical
and biological situations, the configuration (phase) space X is (partially)
shared by identical or closely related representatives of several species. For
example, an ideal monoatomic gas can be represented by a collection of
points in R3 , where atoms do not interact with each other. More generally,
a gas is constituted by several species of molecules with internal (classical or
quantum) degrees of freedom and with mutual hard-core interactions. The
system becomes more interesting if we allow chemical reactions leading to
the global configuration (phase) space of fluctuating dimension. In biology,
one speaks of populations of bacteria in solution where each bacterium in
itself makes a statistical dynamical system.
The (classical) symmetry of a population under permutations of individuals within a given species, can be encompassed by the language of population
states. A population state over X is a measure π on X where π(Y ), Y ⊂ X,
is interpreted as the number of representatives of a given species in Y . If we
deal with several species, this π should be vector valued, where the dimension
of the range of this measure equals the number of species. Furthermore, if
the species have internal degrees of freedom, the measure may take values
in a more sophisticated category than in R+ or Rn+ . Sometimes, this can be
viewed as a measure of some extension of X. For example, for a diatomic
gas in R3 , the relevant π is the measure on the space of line elements in R3 .
The above discussion can be formally shifted from X to the space Π(X)
of populations (measures) π(X), where one should be aware of the following
issues:
- measures π representing actual populations are of rather special nature.
For example one can limit to integer valued measures or to measures with a
quite small support in X.
- relevant dynamics on Π(X) are not linear, but often the non-linearity is
well localized. For example, if one has an ensemble of identical particles with
hard-core interactions, then the non-linearity occurs only at the collisions.
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- a pertinent definition of fuzzy populations and stochastic populations must
incorporate the specificity of the space Π(X). The classical example is the
Poisson distribution σ on a space X, which is, in the present language, a
stochastic population of points (states) in X. This is a probability measure
on the space Π = ΠZ+ (X) of integer valued measures on X, which satisfies
the Poisson independence axiom: let Yi , with i ∈ I, be open subsets in X and
let mi be non-negative integers. Denote by Pi ∈ Π the subset of measures π
such that π(Yi ) = mi . If the subsets Yi are mutually disjoint, then
σ(Πi ).
σ( Πi ) =
i∈I
i∈I
An essentially equivalent way to express the same idea is by saying that
σ defines a Boltzmann sheaf, that is a sheaf of measurable spaces over X.
(The notion of such a sheaf makes sense since the category of measure spaces
admits fiber products.)
The formalism of stochastic population may look rather farfetched but it
seems necessary for a consistent statistical description of biological systems
where one needs an even more refined language describing the local/global
features of the system and the rate of decay of correlations between individuals in a population. We shall postpone a detailed discussion of this until
another occasion.
Genotype, phenotype and syntactic envelops.
The idea of genotype can be formalized in a variety of ways starting from
the space S of 4 letter sequences, for simplicity of a fixed (large) length N .
Most naively, a genotype is represented by a single sequence S ∈ S; next, it
may be a subset in S; more appropriately, one deals with probability measures on S, interpreted either as random genomes, or as populations. Speaking of populations, one enlarges the setting by allowing sets of populations
and random populations.
It is less clear what are mathematical objects corresponding to the notion
of phenotype. Intuitively, the “mathematical phenotype” should represent an
equivalence class of dynamical systems of the kind seen in the previous section, capturing essential features of their behavior. Eventually, a phenotype
reduces to a point in some space of observables or to a probability measure on
such a space. Furthermore, a phenotype usually appears not as an individual
81
object but rather as a category of these, where the morphisms correspond to
transformations (reductions) of observable quantities.
After having chosen suitable definitions for spaces G of genotypes and P
of phenotypes, one studies the correspondence (is it a map?) from G to P.
The first issue is finding a simple syntactic description of a given genotype
G ∈ G with a prescribed (phenotypical) error bound. We think of this
description as an enlargement of a given sequence or of a sample of sequences,
and call it a syntactic 8-envelop of G, where “syntactic” refers to a chosen
formal language and “8” is the size of the phenotypical error.
As a matter of example, let us briefly indicate syntactic possibilities of
describing probability measures on S. The simplest class of measures is given
by the product measures with weights assigned to the four letters. Such a description is very different from explicitly writing down an individual random
sequence with the distribution law described by the measure. The latter,
for a large N , is much longer than the formal description of the underlying
9
product measure: the product measure on the space {A, T, C, G}3·10 with
four equal weights for occurrences of the letters, is essentially described in the
above line and a half; a description of a point in this space, e.g. the genome
of a human individual, takes at least 107 − 108 such lines. On the other hand,
the phenotypical effects of two quite different random strings with identical
underlying probability measure will be most similar if not identical: no living
system run by such a genotype exists or can exists.
Fibered measures. Next class of measures on the space S = SN of words
of length N is constituted by fibered measures defined by four sequences
of
pi =
functions pi : Sj → [0, 1], where i = 1, . . . , 4, j = 1 . . . N − 1 and
1. Given the functions pi , one obtains fibered measures µj (Sj ) where the
corresponding random sequences are defined by the following rule: the j-th
letter in a word is chosen on random with the probability weight equal to the
value of pi evaluated on the previous j −1 letters of the word. The complexity
of such a measure is understood as the complexity of the functions pi , where
the many options for measuring complexity are offered by the traditional
computational complexity theory. (There are many variations of this class
of measures, essentially due to different ways (or precisions) of ordering the
letters, dividing them into blocks etc.)
Parametric measures. Given a measure µ0 on SM and a parameterization
map ψ : SM → SN , we have the push-forward measure µ = ψ∗ (µ0 ) on
82
SN . The complexity of µ can be measured by the sum of the complexities
of µ0 and of the map ψ, where the latter is understood as for a boolean
function. (One could embrace fibered and parametric measures in a single
more general definition. Also, one could generalize parametric measures by
allowing random maps from SM to SN , that are linear maps between the
spaces of probability measures of the corresponding sequence spaces.)
The upshot of the two above constructive definitions is the possibility to
introduce a notion of complexity (in fact several such notions) for measures
on SN .
Questions. Given a measure µ0 (e.g. coming from experiments), a phenomenological map f : SN → P, e.g. for P = Rk , and an 8 > 0, when and
how can one evaluate (from above and from below) the complexity of a measure µ such that dist(f∗ (µ), f∗ (µ0 )) ≤ 8, for a suitable notion of distance on
the space of measures on P 54 . (If P = Rk , then one can use " µ(f ) − µ0 (f ) "
for example.)
What happens to complexity of measures and approximations, when SN
converges to the Cantor set (of infinite sequences), for N → ∞?
Both questions have relativized counterparts, where the complexity is
measured modulo additional information. For example, 3 · 109 letters making
the individual human genome can be reduced to mere 106 − 107 symbols
(where the variation in the coding parts, called single nucleotide variations
or SNP for short, allow ≈ 104 variations).
Remark. The above suggests a (possibly useful) interpolation between
two foundations of the probability theory: the Kolmogorov complexity and
the traditional measure theoretic approach (also due to Kolmogorov).
Remark. The phenotypical map f is not explicitly given in any realistic
situation, and can be treated, to a certain extent, as the above “experimental
measure” µ0 55 .
54
Mathematical statistics is concerned with finding rather explicitly described measures
µ, such as Gaussian measures, approximating an “experimental” measure.
55
One can speculate that this map could be derived from fundamental physical laws if
we had had an infinite computational power at our disposal. Since this is unavailable,
one searches for particular patterns and regularities in specific maps and measures coming
from (biological) experiments and allowing a feasible description of these objects.
83
8
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Tables and Figures:
Some of these have been adapted from the textbooks cited above, where the
reader can find more detailed information.
86